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Page iii
PREFACE
Algebra 2 and Trigonometry is a new text for a course in intermediate algebra and
trigonometry that continues the approach that has made Amsco a leader in presenting mathematics in a modern, integrated manner. Over the last decade, this
approach has undergone numerous changes and refinements to keep pace with
ever-changing technology.
This textbook is the final book in the three-part series in which Amsco parallels
the integrated approach to the teaching of high school mathematics promoted by
the National Council of Teachers of Mathematics in its Principles and Standards for
School Mathematics and mandated by the New York State Board of Regents in the
Mathematics Core Curriculum. The text presents a range of materials and explanations that are guidelines for achieving a high level of excellence in their understanding of mathematics.
In this book:
✔
The real numbers are reviewed and the understanding of operations with irrational numbers, particularly radicals, is expanded.
✔
The graphing calculator continues to be used as a routine tool in the study of
mathematics. Its use enables the student to solve problems that require computation
that more realistically reflects the real world. The use of the calculator replaces the
need for tables in the study of trigonometry and logarithms.
✔
Coordinate geometry continues to be an integral part of the visualization of
algebraic and trigonometric relationships.
✔
Functions represent a unifying concept throughout. The algebraic functions
introduced in Integrated Algebra 1 are reviewed, and exponential, logarithmic, and
trigonometric functions are presented.
✔
Algebraic skills from Integrated Algebra 1 are maintained, strengthened, and
expanded as both a holistic approach to mathematics and as a bridge to advanced
studies.
✔
Statistics includes the use of the graphing calculator to reexamine range, quartiles, and interquartile range, to introduce measures of dispersion such as variance
and standard deviation, and to determine the curve that best represents a set of
bivariate data.
iii
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PREFACE
✔
Integration of geometry, algebra, trigonometry, statistics, and other branches of
mathematics begun in Integrated Algebra 1 and Geometry is continued and further
expanded.
✔
Exercises are divided into three categories. Writing About Mathematics encourages the student to reflect on and justify mathematical conjectures, to discover
counterexamples, and to express mathematical ideas in his or her own words.
Developing Skills provides routine practice exercises that enable the student and
teacher to evaluate the student’s ability to both manipulate mathematical symbols
and understand mathematical relationships. Applying Skills provides exercises in
which the new ideas of each section, together with previously learned skills, are used
to solve problems that reflect real-life situations.
✔
Problem solving, a primary goal of all learning standards, is emphasized
throughout the text. Students are challenged to apply what has been learned to the
solution of both routine and non-routine problems.
✔
Enrichment is stressed both in the text and in the Teacher’s Manual where
many suggestion are given for teaching strategies and alternative assessment. The
Manual provides opportunities for extended tasks and hands-on activities.
Reproducible Enrichment Activities that challenge students to explore topics in
greater depth are provided in each chapter of the Manual.
In this text, the real number system is expanded to include the complex numbers, and algebraic, exponential, logarithmic, and trigonometric functions are investigated. The student is helped to understand the many branches of mathematics, to
appreciate the common threads that link these branches, and to recognize their
interdependence.
The intent of the author is to make this book of greatest service to the average
student through detailed explanations and multiple examples. Each section provides
careful step-by-step procedures for solving routine exercises as well as the nonroutine applications of the material. Sufficient enrichment material is included to
challenge students of all abilities.
Specifically:
✔
Concepts are carefully developed using appropriate language and mathematical symbolism. General principles are stated clearly and concisely.
✔
Numerous examples serve as models for students with detailed explanations of
the mathematical concepts that underlie the solution. Alternative approaches are
suggested where appropriate.
✔
Varied and carefully graded exercises are given in abundance to develop skills
and to encourage the application of those skills. Additional enrichment materials
challenge the most capable students.
This text is offered so that teachers may effectively continue to help students to
comprehend, master, and enjoy mathematics as they progress in their education.
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CONTENTS
Chapter 1
THE INTEGERS
1-1
1-2
1-3
1-4
1-5
1-6
1-7
1-8
1
Whole Numbers, Integers, and the Number Line
Writing and Solving Number Sentences
Adding Polynomials
Solving Absolute Value Equations and Inequalities
Multiplying Polynomials
Factoring Polynomials
Quadratic Equations with Integral Roots
Quadratic Inequalities
Chapter Summary
Vocabulary
Review Exercises
2
5
9
13
17
22
27
30
35
36
37
Chapter 2
THE RATIONAL NUMBERS
39
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
40
44
48
53
57
61
64
70
74
74
75
77
Rational Numbers
Simplifying Rational Expressions
Multiplying and Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Ratio and Proportion
Complex Rational Expressions
Solving Rational Equations
Solving Rational Inequalities
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
Chapter 3
REAL NUMBERS AND RADICALS
3-1
3-2
3-3
The Real Numbers and Absolute Value
Roots and Radicals
Simplifying Radicals
79
80
84
88
v
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CONTENTS
3-4
3-5
3-6
3-7
3-8
Adding and Subtracting Radicals
Multiplying Radicals
Dividing Radicals
Rationalizing a Denominator
Solving Radical Equations
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
94
98
102
104
108
113
114
114
117
Chapter 4
RELATIONS AND FUNCTIONS
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
4-10
Relations and Functions
Function Notation
Linear Functions and Direct Variation
Absolute Value Functions
Polynomial Functions
The Algebra of Functions
Composition of Functions
Inverse Functions
Circles
Inverse Variation
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
119
120
127
130
136
140
149
155
160
167
174
178
180
180
184
Chapter 5
QUADRATIC FUNCTIONS AND COMPLEX NUMBERS
5-1
5-2
5-3
5-4
5-5
5-6
5-7
5-8
5-9
Real Roots of a Quadratic Equation
The Quadratic Formula
The Discriminant
The Complex Numbers
Operations with Complex Numbers
Complex Roots of a Quadratic Equation
Sum and Product of the Roots of a Quadratic Equation
Solving Higher Degree Polynomial Equations
Solutions of Systems of Equations and Inequalities
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
186
187
193
198
203
209
217
219
224
229
239
240
241
244
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CONTENTS
vii
Chapter 6
SEQUENCES AND SERIES
6-1
6-2
6-3
6-4
6-5
6-6
6-7
Sequences
Arithmetic Sequences
Sigma Notation
Arithmetic Series
Geometric Sequences
Geometric Series
Infinite Series
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
247
248
252
257
262
266
270
273
279
280
280
283
Chapter 7
EXPONENTIAL FUNCTIONS
7-1
7-2
7-3
7-4
7-5
7-6
7-7
Laws of Exponents
Zero and Negative Exponents
Fractional Exponents
Exponential Functions and Their Graphs
Solving Equations Involving Exponents
Solving Exponential Equations
Applications of Exponential Functions
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
286
287
289
293
298
304
306
308
314
315
315
316
Chapter 8
LOGARITHMIC FUNCTIONS
8-1
8-2
8-3
8-4
8-5
8-6
8-7
Inverse of an Exponential Function
Logarithmic Form of an Exponential Equation
Logarithmic Relationships
Common Logarithms
Natural Logarithms
Exponential Equations
Logarithmic Equations
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
319
320
324
327
332
336
340
344
347
347
348
351
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CONTENTS
Chapter 9
TRIGONOMETRIC FUNCTIONS
9-1
9-2
9-3
9-4
9-5
9-6
9-7
9-8
Trigonometry of the Right Triangle
Angles and Arcs as Rotations
The Unit Circle, Sine, and Cosine
The Tangent Function
The Reciprocal Trigonometric Functions
Function Values of Special Angles
Function Values from the Calculator
Reference Angles and the Calculator
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
353
354
357
362
368
374
378
381
386
392
394
394
396
Chapter 10
MORE TRIGONOMETRIC FUNCTIONS
10-1
10-2
10-3
10-4
10-5
10-6
Radian Measure
Trigonometric Function Values and Radian Measure
Pythagorean Identities
Domain and Range of Trigonometric Functions
Inverse Trigonometric Functions
Cofunctions
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
399
400
406
411
414
419
425
428
430
430
431
Chapter 11
GRAPHS OF TRIGONOMETRIC FUNCTIONS
11-1
11-2
11-3
11-4
11-5
11-6
11-7
11-8
Graph of the Sine Function
Graph of the Cosine Function
Amplitude, Period, and Phase Shift
Writing the Equation of a Sine or Cosine Graph
Graph of the Tangent Function
Graphs of the Reciprocal Functions
Graphs of Inverse Trigonometric Functions
Sketching Trigonometric Graphs
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
434
435
442
447
455
460
463
468
472
475
476
476
479
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CONTENTS
ix
Chapter 12
TRIGONOMETRIC IDENTITIES
12-1
12-2
12-3
12-4
12-5
12-6
12-7
12-8
Basic Identities
Proving an Identity
Cosine (A 2 B)
Cosine (A 1 B)
Sine (A 2 B) and Sine (A 1 B)
Tangent (A 2 B) and Tangent (A 1 B)
Functions of 2A
Functions of 12A
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
482
483
485
488
493
496
500
504
508
513
514
514
515
Chapter 13
TRIGONOMETRIC EQUATIONS
13-1
13-2
13-3
13-4
13-5
First-Degree Trigonometric Equations
Using Factoring to Solve Trigonometric Equations
Using the Quadratic Formula to Solve Trigonometric Equations
Using Substitution to Solve Trigonometric Equations
Involving More Than One Function
Using Substitution to Solve Trigonometric Equations
Involving Different Angle Measures
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
518
519
526
530
534
538
542
542
543
545
Chapter 14
TRIGONOMETRIC APPLICATIONS
14-1
14-2
14-3
14-4
14-5
14-6
14-7
Similar Triangles
Law of Cosines
Using the Law of Cosines to Find Angle Measure
Area of a Triangle
Law of Sines
The Ambiguous Case
Solving Triangles
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
547
548
552
557
559
564
569
575
581
582
582
585
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CONTENTS
Chapter 15
STATISTICS
15-1
15-2
15-3
15-4
15-5
15-6
15-7
15-8
15-9
15-10
587
Univariate Statistics
Measures of Central Tendency
Measures of Central Tendency for Grouped Data
Measures of Dispersion
Variance and Standard Deviation
Normal Distribution
Bivariate Statistics
Correlation Coefficient
Non-Linear Regression
Interpolation and Extrapolation
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
588
596
605
614
619
628
634
641
647
655
662
664
664
669
Chapter 16
PROBABILITY AND THE BINOMIAL THEOREM
16-1
16-2
16-3
16-4
16-5
16-6
INDEX
The Counting Principle
Permutations and Combinations
Probability
Probability with Two Outcomes
Binomial Probability and the Normal Curve
The Binomial Theorem
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
672
673
678
687
695
701
708
711
713
713
715
718
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Page 1
CHAPTER
THE
INTEGERS
In golf tournaments, a player’s standing after each
hole is often recorded on the leaderboard as the number of strokes above or below a standard for that hole
called a par. A player’s standing is a positive number if
the number of strokes used was greater than par and
a negative number if the number of strokes used was
less than par. For example, if par for the first hole is 4
strokes and a player uses only 3, the player’s standing
after playing the first hole is 21.
Rosie Barbi is playing in an amateur tournament.
Her standing is recorded as 2 below par (22) after sixteen holes. She shoots 2 below par on the seventeenth
hole and 1 above par on the eighteenth. What is
Rosie’s standing after eighteen holes? Nancy Taylor,
who is her closest opponent, has a standing of 1 below
par (21) after sixteen holes, shoots 1 below par on
the seventeenth hole and 1 below par on the eighteenth.What is Nancy’s standing after eighteen holes?
In this chapter, we will review the set of integers
and the way in which the integers are used in algebraic
expressions, equations, and inequalities.
1
CHAPTER
TABLE OF CONTENTS
1-1 Whole Numbers, Integers,
and the Number Line
1-2 Writing and Solving Number
Sentences
1-3 Adding Polynomials
1-4 Solving Absolute Value
Equations and Inequalities
1-5 Multiplying Polynomials
1-6 Factoring Polynomials
1-7 Quadratic Equations with
Integral Roots
1-8 Quadratic Inequalities
Chapter Summary
Vocabulary
Review Exercises
1
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The Integers
1-1 WHOLE NUMBERS, INTEGERS, AND THE NUMBER LINE
The first numbers that we learned as children and probably the first numbers
used by humankind are the natural numbers. Most of us began our journey of
discovery of the mathematical world by counting, the process that lists, in order,
the names of the natural numbers or the counting numbers. When we combine
the natural numbers with the number 0, we form the set of whole numbers:
{0, 1, 2, 3, 4, 5, 6, . . . }
These numbers can be displayed as points on the number line:
0
1
2
3
4
5
6
7
8
9
The number line shows us the order of the whole numbers; 5 is to the right
of 2 on the number line because 5 2, and 3 is to the left of 8 on the number
line because 3 8. The number 0 is the smallest whole number. There is no largest whole number.
The temperature on a winter day may be two degrees above zero or two
degrees below zero. The altitude of the highest point in North America is 20,320
feet above sea level and of the lowest point is 282 feet below sea level. We represent numbers less than zero by extending the number line to the left of zero,
that is, to numbers that are less than zero, and by assigning to every whole number a an opposite, 2a, such that a 1 (2a) 5 0.
DEFINITION
The opposite or additive inverse of a is 2a, the number such that
a 1 (2a) 5 0.
The set of integers is the union of the set of whole numbers and their opposites. The set of non-zero whole numbers is the positive integers and the opposites of the positive integers are the negative integers.
6 5 4 3 2 1
0
1
2
3
4
5
6
Let a, b, and c represent elements of the set of integers. Under the operation of addition, the following properties are true:
1. Addition is closed:
a 1 b is an integer
2. Addition is commutative:
a1b5b1a
3. Addition is associative:
(a 1 b) 1 c 5 a 1 (b 1 c)
4. Addition has an identity element, 0:
a105a
5. Every integer has an inverse:
a 1 (2a) 5 0
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Whole Numbers, Integers, and the Number Line
3
We say that the integers form a commutative group under addition because
the five properties listed above are true for the set of integers.
Subtraction
DEFINITION
a 2 b 5 c if and only if b 1 c 5 a.
Solve the equation b 1 c 5 a for c:
b1c5a
2b 1 b 1 c 5 a 1 (2b)
c 5 a 1 (2b)
Therefore, a 2 b = a 1 (2b).
Absolute Value
A number, a, and its opposite, 2a, are the same distance from zero on the number line. When that distance is written as a positive number, it is called the
absolute value of a.
• If a 0, then a 5 a 2 0 5 a
• If a 0, then a 5 0 2 a 5 2a
Note: When a 0, a is a negative number and its opposite, 2a, is a positive
number.
For instance, 5 0. Therefore, 5 5 5 2 0 5 5.
25 0. Therefore, –5 5 0 2 (25) 5 5.
We can also say that a 5 –a 5 a or 2a, whichever is positive.
EXAMPLE 1
Show that the opposite of 2b is b.
Solution The opposite of b, 2b, is the number such that b 1 (2b) 5 0.
Since addition is commutative, b 1 (2b) 5 (2b) 1 b 5 0.
The opposite of 2b is the number such that (2b) 1 b 5 0. Therefore, the
opposite of 2b is b.
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The Integers
Exercises
Writing About Mathematics
1. Tina is three years old and knows how to count. Explain how you would show Tina that
3 1 2 5 5.
2. Greg said that a 2 b 5 b 2 a. Do you agree with Greg? Explain why or why not.
Developing Skills
In 3–14, find the value of each given expression.
3. 6
4. 212
5. 8 2 3
6. 3 2 8
7. 5 1 (212)
8. 212 1 (2(25))
9. 4 2 6 1 (22)
12. 8 2 22 2 2
10. 8 1 (10 2 18)
11. 3 2 3
13. 2(22 1 3)
14. 4 2 3 1 21
In 15–18, use the definition of subtraction to write each subtraction as a sum.
15. 8 2 5 5 3
16. 7 2 (22) 5 9
17. 22 2 5 5 27
18. 28 2 (25) 5 23
19. Two distinct points on the number line represent the numbers a and b.
If 5 2 a 5 5 2 b 5 6, what are the values of a and b?
Applying Skills
In 20–22, Mrs. Menendez uses computer software to record her checking account balance. Each time
that she makes an entry, the amount that she enters is added to her balance.
20. If she writes a check for $20, how should she enter this amount?
21. Mrs. Menendez had a balance of $52 in her checking account and wrote a check for $75.
a. How should she enter the $75?
b. How should her new balance be recorded?
22. After writing the $75 check, Mrs. Menendez realized that she would be overdrawn when the
check was paid by the bank so she transferred $100 from her savings account to her checking account. How should the $100 be entered in her computer program?
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Writing and Solving Number Sentences
5
1-2 WRITING AND SOLVING NUMBER SENTENCES
Equations
A sentence that involves numerical quantities can often be written in the symbols of algebra as an equation. For example, let x represent any number. Then
the sentence “Three less than twice a number is 15” can be written as:
2x 2 3 5 15
When we translate from one language to another, word order often must be
changed in accordance with the rules of the language into which we are translating. Here we must change the word order for “three less than twice a number” to match the correct order of operations.
The domain is the set of numbers that can replace the variable in an algebraic expression. A number from the domain that makes an equation true is a
solution or root of the equation. We can find the solution of an equation by writing a sequence of equivalent equations, or equations that have the same solution set, until we arrive at an equation whose solution set is evident. We find
equivalent equations by changing both sides of the given equation in the same
way. To do this, we use the following properties of equality:
Properties of Equality
• Addition Property of Equality: If equals are added to equals, the sums
are equal.
• Subtraction Property of Equality: If equals are subtracted from
equals, the differences are equal.
• Multiplication Property of Equality: If equals are multiplied by equals,
the products are equal.
• Division Property of Equality: If equals are divided by non-zero
equals, the quotients are equal.
On the left side of the equation 2x 2 3 5 15, the variable is multiplied by 2
and then 3 is subtracted from the product. We will simplify the left side of the
equation by “undoing” these operations in reverse order, that is, we will first add
3 and then divide by 2. We can check that the number we found is a root of the
given equation by showing that when it replaces x, it gives us a correct statement
of equality.
2x 2 3 5 15
2x 2 3 1 3 5 15 1 3
2x 5 18
x59
Check
2x 2 3 5 15
?
2(9) 2 3 5 15
15 5 15 ✔
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The Integers
Often the definition of a mathematical term or a formula is needed to write
an equation as the following example demonstrates:
EXAMPLE 1
Let A be an angle such that the complement of A is 6 more than twice the
measure of A. Find the measure of A and its complement.
Solution To write an equation to find A, we must know that two angles are complements if the sum of their measures is 90°.
Let x 5 the measure of A.
Then 2x 1 6 5 the measure of the complement of A.
The sum of the measures of an angle and of its complement is 90.
x 1 2x 1 6 5 90
3x 1 6 5 90
3x
5 84
x
5 28
2x 1 6 5 2(28) 1 6 5 62
Therefore, the measure of A is 28 and the measure of its complement is 62.
Check The sum of the measures of A and its complement is 28 1 62 or 90. ✔
Answer mA 5 28; the measure of the complement of A is 62.
EXAMPLE 2
Find the solution of the following equation: 6x 2 3 5 15.
Solution Since 15 5 215 5 15, the algebraic expression 6x 2 3 can be equal to 15 or
to 215.
6x 2 3 5 15
6x 2 3 1 3 5 15 1 3
or
6x 2 3 5 215
6x 2 3 1 3 5 215 1 3
6x 5 18
6x 5 212
x53
x 5 22
Check: x 5 3
6x 2 3 5 15
?
6(3) 2 3 5 15
15 5 15 ✔
Answer The solution set is {3, 22}.
Check: x 5 22
6x 2 3 5 15
?
6(22) 2 3 5 15
215 5 15 ✔
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Writing and Solving Number Sentences
7
Inequalities
A number sentence can often be an inequality. To find the solution set of an
inequality, we use methods similar to those that we use to solve equations. We
need the following two properties of inequality:
Properties of Inequality
• Addition and Subtraction Property of Inequality: If equals are added
to or subtracted from unequals, the sums or differences are unequal in
the same order.
• Multiplication and Division Property of Inequality: If unequals are
multiplied or divided by positive equals, the products or quotients are
unequal in the same order. If unequals are multiplied or divided by
negative equals, the products or quotients are unequal in the opposite
order.
EXAMPLE 3
Find all positive integers that are solutions of the inequality 4n 1 7 27.
Solution We solve this inequality by using a procedure similar to that used for solving
an equation.
4n 1 7 , 27
4n 1 7 1 (27) , 27 1 (27)
4n , 20
n,5
Since n is a positive integer, the solution set is {1, 2, 3, 4}. Answer
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The Integers
EXAMPLE 4
Polly has $210 in her checking account. After writing a check for tickets to a concert, she has less than $140 in her account but she is not overdrawn. If each
ticket cost $35, how many tickets could she have bought?
Solution Let x 5 the number of tickets.
The cost of x tickets, 35x, will be subtracted from $210, the amount in her
checking account. Since she is not overdrawn after writing the check, her balance is at least 0 and less than $140.
0 210 2 35x 140
–210
–210
–210 2210
235
6 –210
Add 2210 to each member of the inequality.
2 35x 270
235x
235
270
235
x2
Divide each member of the inequality by 235.
Note that dividing by a negative number
reverses the order of the inequality.
Polly bought more than 2 tickets but at most 6.
Answer Polly bought 3, 4, 5, or 6 tickets.
Exercises
Writing About Mathematics
1. Explain why the solution set of the equation 12 2 x 5 15 is the empty set.
2. Are 24x 12 and x 23 equivalent inequalities? Justify your answer.
Developing Skills
In 3–17, solve each equation or inequality. Each solution is an integer.
3. 5x 1 4 5 39
4. 7x 1 18 5 39
5. 3b 1 18 5 12
6. 12 2 3y 5 18
7. 9a 2 7 5 29
8. 13 2 x 5 15
9. 2x 1 4 5 22
10. 3 2 y 5 8
11. 4a 2 12 5 16
12. 2x 1 3| 2 8 5 15
13. 7a 1 3 17
14. 9 2 2b 1
15. 3 4x 2 1 11
16. 0 , x 2 3 , 4
17. 5 $ 4b 1 9 $ 17
Applying Skills
In 18–23, write and solve an equation or an inequality to solve the problem.
18. Peter had 156 cents in coins. After he bought 3 packs of gum he had no more than 9 cents
left. What is the minimum cost of a pack of gum?
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Adding Polynomials
9
19. In an algebra class, 3 students are working on a special project and the remaining students
are working in groups of five. If there are 18 students in class, how many groups of five are
there?
20. Andy paid a reservation fee of $8 plus $12 a night to board her cat while she was on vacation. If Andy paid $80 to board her cat, how many nights was Andy on vacation?
21. At a parking garage, parking costs $5 for the first hour and $3 for each additional hour or
part of an hour. Mr. Kanesha paid $44 for parking on Monday. For how many hours did Mr.
Kanesha park his car?
22. Kim wants to buy an azalea plant for $19 and some delphinium plants for $5 each. She wants
to spend less than $49 for the plants. At most how many delphinium plants can she buy?
23. To prepare for a tennis match and have enough time for schoolwork, Priscilla can practice
no more than 14 hours.. If she practices the same length of time on Monday through Friday,
and then spends 4 hours on Saturday, what is the most time she can practice on Wednesday?
1-3 ADDING POLYNOMIALS
A monomial is a constant, a variable, or the product of constants and variables.
Each algebraic expression, 3, a, ab, 22a2, is a monomial.
Exponent
➛
3a4
➛
➛
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Coefficient
Base
A polynomial is the sum of monomials. Each monomial is a term of the
polynomial. The expressions 3a2 1 7a 2 2 is a polynomial over the set of integers since all of the numerical coefficients are integers. For any integral value of
a, 3a2 1 7a 2 2 has an integral value. For example, if a 5 22, then:
3a2 1 7a 2 2 5 3(22)2 1 7(22) 22
5 3(4) 1 7(22) 2 2
5 12 2 14 2 2
5 24
The same properties that are true for integers are true for polynomials: we
can use the commutative, associative, and distributive properties when working
with polynomials. For example:
(3a2 1 5a) 1 (6 2 7a) 5 (3a2 1 5a) 1 (27a 1 6)
Commutative Property
2
Associative Property
2
Distributive Property
5 3a 1 (5a 2 7a) 1 6
5 3a 1 (5 2 7)a 1 6
2
5 3a 2 2a 1 6
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Note: When the two polynomials are added, the two terms that have the same
power of the same variable factor are combined into a single term.
Two terms that have the same variable and exponent or are both numbers
are called similar terms or like terms. The sum of similar terms is a monomial.
3a2 1 5a2
27ab 1 3ab
x3 1 4x3
5 (3 1 5)a2
5 (27 1 3)ab
5 (1 1 4)x3
5 8a2
5 24ab
5 5x3
Two monomials that are not similar terms cannot be combined. For example, 4x3 and 3x2 are not similar terms and the sum 4x3 1 3x2 is not a monomial.
A polynomial in simplest form that has two terms is a binomial. A polynomial
in simplest form that has three terms is a trinomial.
Solving Equations and Inequalities
An equation or inequality often has a variable term on both sides. To solve such
an equation or inequality, we must first write an equivalent equation or inequality with the variable on only one side.
For example, to solve the inequality 5x 2 7 3x 1 9, we will first write an
equivalent inequality that does not have a variable in the right side. Add the
opposite of 3x, 23x, to both sides. The terms 3x and 23x are similar terms whose
sum is 0.
5x 2 7 3x 1 9
23x 1 5x 2 7 23x 1 3x 1 9
2x 2 7 9
2x 2 7 1 7 9 1 7
2x 16
x8
Add 23x, the opposite of 3x, to both sides.
23x 1 3x 5 (23 1 3)x 5 0x 5 0
Add 7, the opposite of 27, to both sides.
Divide both sides by 2. Dividing by a
positive does not reverse the inequality.
If x is an integer, then the solution set is {9, 10, 11, 12, 13, . . . }.
EXAMPLE 1
a. Find the sum of x3 2 5x 1 9 and x 2 3x3.
b. Find the value of each of the given polynomials and the value of their sum
when x 5 24.
Solution a. The commutative and associative properties allow us to change the order
and the grouping of the terms.
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11
(x3 2 5x 1 9) 1 (x 2 3x3) 5 (x3 2 3x3) 1 (25x 1 x) 1 9
5 (1 2 3)x3 1 (25 1 1)x 1 9
5 22x3 2 4x 1 9 Answer
b.
x3 2 5x 1 9
x 2 3x3
22x3 2 4x 1 9
5 (24) 3 2 5(24) 1 9
5 (24) 2 3(24) 3
5 22(24) 3 2 4(24) 1 9
5 264 1 20 1 9
5 24 1 192
5 128 1 16 1 9
5 235 Answer
5 188 Answer
5 153 Answer
EXAMPLE 2
Subtract (3b4 1 b 1 3) from (b4 2 5b 1 3) and write the difference as a polynomial in simplest form.
Solution Subtract (3b4 1 b 1 3) from (b4 2 5b 1 3) by adding the opposite of
(3b4 1 b 1 3) to (b4 2 5b 1 3).
(b4 2 5b 1 3) 2 (3b4 1 b 1 3) 5 (b4 2 5b 1 3) 1 (23b4 2 b 2 3)
5 (b4 2 3b4) 1 (25b 2 b) 1 (3 2 3)
5 22b4 2 6b Answer
EXAMPLE 3
Pam is three times as old as Jody. In five years, Pam will be twice as old as Jody.
How old are Pam and Jody now?
Solution
Let x 5 Jody’s age now
3x 5 Pam’s age now
x 1 5 5 Jody’s age in 5 years
3x 1 5 5 Pam’s age in 5 years
Pam’s age in 5 years will be twice Jody’s age in 5 years.
3x 1 5
5
2(x 1 5)
3x 1 5
5
2x 1 10
22x 1 3x 1 5
x15
Answer Jody is 5 and Pam is 15.
5 2 2x 1 2x 1 10
0
5
x 1 5 2 55
10 2 5
x
5
5
3x
5
15
1 10
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Exercises
Writing About Mathematics
1. Danielle said that there is no integer that makes the inequality 2x 1 1 x true. Do you
agree with Danielle? Explain your answer.
2. A binomial is a polynomial with two terms and a trinomial is a polynomial with three terms.
Jess said that the sum of a trinomial and binomial is always a trinomial. Do you agree with
Jess? Justify your answer.
Developing Skills
In 3–12, write the sum or difference of the given polynomials in simplest form.
3. (3y 2 5) 1 (2y 2 8)
4. (x2 1 3x 2 2) 1 (4x2 2 2x 1 3)
5. (4x2 2 3x 2 7) 1 (3x2 2 2x 1 3)
6. (2x2 1 5x 1 8) 1 (x2 2 2x 2 8)
7. (a2b2 2 ab 1 5) 1 (a2b2 1 ab 2 3)
8. (7b2 2 2b 1 3) 2 (3b2 1 8b 1 3)
9. (3 1 2b 1 b2) 2 (9 1 5b 1 b2)
11. (y2 2 y 2 7) 1 (3 2 2y 1 3y2)
10. (4x2 2 3x 2 5) 2 (3x2 2 10x 1 3)
12. (2a4 2 5a2 2 1) 1 (a3 1 a)
In 13–22, solve each equation or inequality. Each solution is an integer.
13. 7x 1 5 5 4x 1 23
14. y 1 12 5 5y 2 4
15. 7 2 2a 5 3a 1 32
16. 12 1 6b 5 2b
17. 2x 1 3 x 1 15
18. 5y 2 1 2y 1 5
19. 9y 1 2 7y
20. 14c 80 2 6c
21. (b 2 1) 2 (3b 2 4) 5 b
22. 23 2 2x $ 12 1 x
Applying Skills
23. An online music store is having a sale. Any song costs 75 cents and any ringtone costs
50 cents. Emma can buy 6 songs and 2 audiobooks for the same price as 5 ringtones and
3 audiobooks. What is the cost of an audiobook?
24. The length of a rectangle is 5 feet more than twice the width.
a. If x represents the width of the rectangle, represent the perimeter of the rectangle in
terms of x.
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13
b. If the perimeter of the rectangle is 2 feet more than eight times the width of the rectangle, find the dimensions of the rectangle.
25. On his trip to work each day, Brady pays the same toll, using either all quarters or all dimes.
If the number of dimes needed for the toll is 3 more than the number of quarters, what is
the toll?
1-4 SOLVING ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
Absolute Value Equations
We know that if a is a positive number, then a 5 a and that 2a 5 a. For example, if x 5 3, then x 5 3 or x 5 23 because 3 5 3 and 23 5 3. We can use
these facts to solve an absolute value equation, that is, an equation containing
the absolute value of a variable.
For instance, solve 2x 2 3 5 17. We know that 17 5 17 and 217 5 17.
Therefore 2x 2 3 can equal 17 or it can equal 217.
2x 2 3 5 17
or
2x 2 3 1 3 5 17 1 3
2x 2 3 5 217
2x 2 3 1 3 5 217 1 3
2x 5 20
2x 5 214
x 5 10
x 5 27
The solution set of 2x 2 3 5 17 is {27, 10}.
In order to solve an absolute value equation, we must first isolate the
absolute value expression. For instance, to solve 4a 1 2 1 7 5 21, we
must first add 27 to each side of the equation to isolate the absolute value
expression.
4a 1 2 1 7 5 21
4a 1 2 1 7 2 7 5 21 2 7
4a 1 2 5 14
Now we can consider the two possible cases: 4a 1 2 5 14 or 4a 1 2 5 214
4a 1 2 5 14
4a 1 2 2 2 5 14 2 2
or
4a 1 2 5 214
4a 1 2 2 2 5 214 2 2
4a 5 12
4a 5 216
a53
a 5 24
The solution set of 4a 1 2 1 7 5 21 is {24, 3}.
Note that the solution sets of the equations x 1 3 5 25 and x 1 3 1 5 5 2
are the empty set because absolute value is always positive or zero.
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EXAMPLE 1
Find the solution of the following equation: 4x 2 2 5 10.
Solution Since 10 5 210 5 10, the algebraic expression 4x 2 2 can be equal to 10 or
to 210.
4x 2 2 5 10
or
4x 2 2 5 210
4x 2 2 1 2 5 10 1 2
4x 2 2 1 2 5 210 1 2
4x 5 12
4x 5 28
x53
x 5 22
Check: x 5 3
4x 2 2 5 10
Check: x 5 22
4x 2 2 5 10
?
?
4(3) 2 2 5 10
4(22) 2 2 5 10
10 5 10 ✔
210 5 10 ✔
Answer The solution set is {3, 22}.
Absolute Value Inequalities
For any two given algebraic expressions, a and b, three relationships are possible: a 5 b, a b, or a b. We can use this fact to solve an absolute value
inequality (an inequality containing the absolute value of a variable). For example, we know that for the algebraic expressions x 2 4 and 3, there are three
possibilities:
CASE 1
x 2 4 5 3
x 2 4 5 23
x51
1 0
1
2
3
or
x2453
or
x57
4
5
6
7
8
Note that the solution set of this inequality consists of the values of x that are 3
units from 4 in either direction.
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Solving Absolute Value Equations and Inequalities
CASE 2
15
x 2 4 3
The solution set of this inequality consists of the values of x that are less
than 3 units from 4 in either direction, that is, x 2 4 is less than 3 and greater
than 23.
x 2 4 23
x1
1 0
1
2
3
and
x243
and
x7
4
5
6
7
8
If x is an integer, the solution set is {2, 3, 4, 5, 6}. Note that these are the integers
between the solutions of x 2 4 5 3.
CASE 3
x 2 4 3
The solution set of this inequality consists of the values of x that are more
than 3 units from 4 in either direction, that is x 2 4 is greater than 3 or less than
23.
x 2 4 23
x1
1 0
1
2
3
or
x243
or
x7
4
5
6
7
8
If x is an integer, the solution set is { . . ., 23, 22, 21, 0, 8, 9, 10, 11, . . .}. Note that
these are the integers that are less than the smaller solution of x 2 4 5 3 and
greater than the larger solution of x 2 4 5 3.
We know that a 5 a if a 0 and a 5 2a if a 0. We can use these relationships to solve inequalities of the form x k and x k.
Solve x k for positive k
Solve x k for positive k
If x 0, x 5 x.
If x 0, x 5 x.
Therefore, x k and 0 x k.
Therefore, x k.
If x 0, x 5 2x.
If x 0, x 5 2x.
Therefore, 2x k or x 2k.
Therefore, 2x k or x 2k.
This can be written 2k x 0.
The solution set of x k is
The solution set of x k is
x 2k or x k.
–k x k.
If x * k for any positive number k, then 2k * x * k.
If x + k for any positive number k, then x + k or x * 2k.
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EXAMPLE 2
Solve for b and list the solution set if b is an integer: 6 2 3b 2 5 4
6 2 3b 2 5 4
Solution (1) Write an equivalent inequality
with only the absolute value on
one side of the inequality:
(2) Use the relationship derived in
this section:
6 2 3b 9
6 2 3b 9 or 6 2 3b 29
If x k for any positive number k,
then x k or x 2k.
(3) Solve each inequality for b:
6 2 3b 9
23b 3
23b
23
,
3
23
6 2 3b 29
23b 215
23b
23
b 21
. 215
23
b5
Answer {. . ., 25, 24, 23, 22, 6, 7, 8, 9, . . .}
Exercises
Writing About Mathematics
1. Explain why the solution of 23b 5 9 is the same as the solution of 3b 5 9.
2. Explain why the solution set of 2x 1 4 1 7 3 is the empty set.
Developing Skills
In 3–14, write the solution set of each equation.
3. x 2 5 5 12
4. x 1 8 5 6
5. 2a 2 5 5 7
6. 5b 2 10 5 25
7. 3x 2 12 5 9
8. 4y 1 2 5 14
9. 35 2 5x 5 10
10. 25a 1 7 5 22
11. 8 1 2b 2 3 5 9
13. 4x 2 12 1 8 5 0
14. 7 2 x 1 2 5 12
12. 2x 2 5 1 2 5 13
In 15–26, solve each inequality and write the solution set if the variable is an element of the set of
integers.
15. x 9
16. y 1 2 7
17. b 1 6 5
18. x 2 3 4
19. y 1 6 13
20. 2b 2 7 9
21. 6 2 3x 15
22. 8 1 4b 0
23. 5 2 b 1 4 9
24. 11 2 2b 2 6 11
25. 6 2 3b 1 4 3
26. 7 2 x 1 2 12
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17
Applying Skills
27. A carpenter is making a part for a desk. The part is to be 256 millimeters wide plus or minus
3 millimeters. This means that the absolute value of the difference between the dimension of
the part and 256 can be no more than 3 millimeters. To the nearest millimeter, what are the
acceptable dimensions of the part?
28. A theater owner knows that to make a profit as well as to comply with fire regulations, the
number of tickets that he sells can differ from 225 by no more than 75. How many tickets
can the theater owner sell in order to make a profit and comply with fire regulations?
29. A cereal bar is listed as containing 200 calories. A laboratory tested a sample of the bars
and found that the actual calorie content varied by as much as 28 calories. Write and solve
an absolute value inequality for the calorie content of the bars.
1-5 MULTIPLYING POLYNOMIALS
We know that the product of any number of equal factors can be written as a
power of that factor. For example:
a 3 a 3 a 3 a 5 a4
In the expression a4, a is the base, 4 is the exponent, and a4 is the power. The
exponent tells us how many times the base, a, is to be used as a factor.
To multiply powers with like bases, keep the same base and add the exponents. For example:
x3 3 x2 5 (x 3 x 3 x) 3 (x 3 x) 5 x5
34 3 35 5 (3 3 3 3 3 3 3) 3 (3 3 3 3 3 3 3 3 3) 5 39
In general:
x a ? x b 5 x a1b
Note that we are not performing the multiplication but simply counting how
many times the base is used as a factor.
Multiplying a Monomial by a Monomial
The product of two monomials is a monomial. We use the associative and commutative properties of multiplication to write the product.
3a2b(2abc) 5 3(2)(a2)(a)(b)(b)(c)
5 6a3b2c
Note: When multiplying (a2)(a) and (b)(b), the exponent of a and of b is 1.
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The square of a monomial is the product of each factor of the monomial
used twice as a factor.
(3ab2) 2
(22x3y) 2
22(x3y) 2
5 (3ab2)(3ab2)
5 (22x3y)(22x3y)
5 22(x3y)(x3y)
5 9a2b4
5 4x6y2
5 22x6y2
Multiplying a Polynomial by a Monomial
To multiply a monomial times a polynomial, we use the distributive property of
multiplication over addition, a(b 1 c) 5 ab 1 ac:
5x(x2 2 3x 1 2)
24(y 2 7)
5 24y 2 4(27)
5 5x(x2) 1 5x(23x) 1 5x(2)
5 24y 1 28
5 5x3 2 15x2 1 10x
Note: The product of a monomial times a polynomial has the same number of
terms as the polynomial.
Multiplying a Polynomial by a Binomial
To multiply a binomial by a polynomial we again use the distributive property of multiplication over addition. First, recall that the distributive property
a(b 1 c) 5 ab 1 ac can be written as:
(b 1 c)a 5 ba 1 ca
Now let us use this form of the distributive property to find the product of two
binomials, for example:
(b 1 c)
(a)
5 b (a)
1c
(a)
(x 1 2)(x 1 5) 5 x(x 1 5) 1 2(x 1 5)
5 x2 1 5x 1 2x 1 10
5 x2 1 7x 1 10
Multiplying two binomials (polynomials with two terms) requires four multiplications. We multiply each term of the first binomial times each term of the
second binomial. The word FOIL helps us to remember the steps needed.
(x 1 4)(x 2 3) 5 x(x 2 3) 1 4(x 2 3)
F
O
I
L
2
5 x 2 3x 1 4x 2 12
5 x2 1 x 2 12
Product of the F irst terms
Product of the O utside terms
Product of the I nside terms
Product of the L ast terms
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19
Multiplying a Polynomial by a Polynomial
To multiply any two polynomials, multiply each term of the first polynomial by
each term of the second. For example:
(a2 1 a 2 3)(2a2 1 3a 2 1)
5 a2(2a2 1 3a 2 1) 1 a(2a2 1 3a 2 1) 2 3(2a2 1 3a 2 1)
5 (2a4 1 3a3 2 a2) 1 (2a3 1 3a2 2 a) 2 (6a2 2 9a 1 3)
5 2a4 1 (3a3 1 2a3) 1 (2a2 1 3a2 2 6a2) 1 (2a 2 9a) 1 3
5 2a4 1 5a3 2 4a2 2 10a 1 3
Note: Since each of the polynomials to be multiplied has 3 terms, there are
3 3 3 or 9 products. After combining similar terms, the polynomial in simplest
form has five terms.
EXAMPLE 1
Write each of the following as a polynomial in simplest form.
a. ab(a2 1 2ab 1 b2)
b. (3x 2 2)(2x 1 5)
c. (y 1 2)(y 2 2)
d. (2a 1 1)(a2 2 2a 2 2)
Solution a. ab(a2 1 2ab 1 b2) 5 a3b 1 2a2b2 1 ab3 Answer
b. (3x 2 2)(2x 1 5) 5 3x(2x 1 5) 2 2(2x 1 5)
5 6x2 1 15x 2 4x 2 10
5 6x2 1 11x 2 10 Answer
c. (y 1 2)(y 2 2) 5 y(y 2 2) 1 2(y 2 2)
5 y2 2 2y 1 2y 2 4
5 y2 2 4 Answer
d. (2a 1 1)(a2 2 2a 2 2) 5 2a(a2 2 2a 2 2) 1 1(a2 2 2a 2 2)
5 2a3 2 4a2 2 4a 1 a2 2 2a 2 2
5 2a3 2 3a2 2 6a 2 2 Answer
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EXAMPLE 2
Write in simplest form: (2b)2 1 5b[2 2 3(b 2 1)]
(2b)2 1 5b[2 2 3(b 2 1)]
Solution (1) Simplify the innermost parentheses first:
5 (2b)2 1 5b[2 2 3b 1 3]
5 (2b)2 1 5b[5 2 3b]
(2) Multiply the terms in the brackets:
5 (2b)2 1 25b 2 15b2
(3) Simplify powers:
5 4b2 1 25b 2 15b2
(4) Add similar terms:
5 25b 2 11b2 Answer
EXAMPLE 3
Solve and check: y(y 1 2) 2 3(y 1 4) 5 y(y 1 1)
Solution (1) Simplify each side of the
equation:
y(y 1 2) 2 3(y 1 4) 5 y(y 1 1)
y2 1 2y 2 3y 2 12 5 y2 1 y
y2 2 y 2 12 5 y2 1 y
(2) Add 2y2 to both sides of the
equation:
2y2 1 y2 2 y 2 12 5 2y2 1 y2 1 y
2y 2 12 5 y
(3) Add y to both sides of the
equation:
y 2 y 2 12 5 y 1 y
212 5 2y
(4) Divide both sides of the
equation by 2:
26 5 y
(5) Check:
y(y 1 2) 2 3(y 1 4) 5 y(y 1 1)
?
–6(26 1 2) 2 3(26 1 4) 5 26(26 1 1)
?
26(24) 2 3(22) 5 –6(25)
?
24 1 6 5 30
30 5 30 ✓
Answer y 5 26
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21
Exercises
Writing About Mathematics
1. Melissa said that (a 1 3)2 5 a2 1 9. Do you agree with Melissa? Justify your answer.
2. If a trinomial is multiplied by a binomial, how many times must you multiply a monomial by
a monomial? Justify your answer.
Developing Skills
In 3–23, perform the indicated operations and write the result in simplest form.
3. 2a5b2(7a3b2)
4. 6c2d(22cd3)
5. (6xy2)2
6. (23c4)2
7. 2(3c4)2
8. 3b(5b 2 4)
2
2
9. 2x y(y 2 2y )
10. (x 1 3)(2x 2 1)
11. (a 2 5)(a 1 4)
12. (3x 1 1)(x 2 2)
13. (a 1 3)(a 2 3)
14. (5b 1 2)(5b 2 2)
15. (a 1 3)
2
16. (3b 2 2)
2
17. (y 2 1)(y2 2 2y 1 1)
18. (2x 1 3)(x2 1 x 2 5)
19. 3a 1 4(2a 2 3)
20. b2 1 b(3b 1 5)
21. 4y(2y 2 3) 2 5(2 2 y)
22. a3(a2 1 3) 2 (a5 1 3a3)
23. (z 2 2)3
In 24–29, solve for the variable and check. Each solution is an integer.
24. (2x 1 1) 1 (4 2 3x) 5 10
25. (3a 1 7) 2 (a 2 1) 5 14
26. 2(b 2 3) 1 3(b 1 4) 5 b 1 14
27. (x 1 3)2 5 (x 2 5)2
28. 4x(x 1 2) 2 x(3 1 4x) 5 2x 1 18
29. y(y 1 2) 2 y(y 2 2) 5 20 2 y
Applying Skills
30. The length of a rectangle is 4 more than twice the width, x. Express the area of the rectangle
in terms of x.
31. The length of the longer leg, a, of a right triangle is 1 centimeter less than the length of the
hypotenuse and the length of the shorter leg, b, is 8 centimeters less than the length of the
hypotenuse.
a. Express a and b in terms of c, the length of the hypotenuse.
b. Express a2 1 b2 as a polynomial in terms of c.
c. Use the Pythagorean Theorem to write a polynomial equal to c2.
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1-6 FACTORING POLYNOMIALS
The factors of a monomial are the numbers and variables whose product is the
monomial. Each of the numbers or variables whose product is the monomial is
a factor of the monomial as well as 1 and any combination of these factors. For
example, the factors of 3a2b are 1, 3, a, and b, as well as 3a, 3b, a2, ab, 3a2, 3ab,
a2b, and 3a2b.
Common Monomial Factor
A polynomial can be written as a monomial times a polynomial if there is at
least one number or variable that is a factor of each term of the polynomial. For
instance:
4a4 2 10a2 5 2a2(2a2 2 5)
Note: 2a2 is the greatest common monomial factor of the terms of the polynomial because 2 is the greatest common factor of 4 and 10 and a2 is the smallest
power of a that occurs in each term of the polynomial.
EXAMPLE 1
Factor:
Answers
2 3
2
a. 12x y 2 15xy 1 9y
5 3y(4x2y2 2 5xy 1 3)
b. a2b3 1 ab2c
5 ab2 (ab 1 c)
c. 2x2 2 8x 1 10
5 2(x2 2 4x 1 5)
Common Binomial Factor
We know that:
5ab
1 3b
If we replace b by (x 1 2) we
can write:
5a(x 1 2) 1 3(x 1 2) 5 (x 1 2)(5a 1 3)
5
b(5a 1 3)
Just as b is the common factor of 5ab 1 3b, (x 1 2) is the common factor of
5a(x 1 2) 1 3(x 1 2). We call (x 1 2) the common binomial factor.
EXAMPLE 2
Find the factors of: a3 1 a2 2 2a 2 2
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23
Solution Find the common factor of the first two terms and the common factor of the
last two terms. Use the sign of the first term of each pair as the sign of the
common factor.
a3 1 a2 2 2a 2 2 5 a2(a 1 1) 2 2(a 1 1)
5 (a 1 1)(a2 2 2) Answer
Note: In the polynomial given in Example 2, the product of the first and last
terms is equal to the product of the two middle terms: a3 (22) 5 a2 (22a).
This relationship will always be true if a polynomial of four terms can be factored into the product of two binomials.
Binomial Factors
We can find the binomial factors of a trinomial, if they exist, by reversing the
process of finding the product of two binomials. For example:
(x 1 3)(x 2 2) 5 x(x 2 2) 1 3(x 2 2)
5 x2 2 2x 1 3x 2 6
5 x2 1 x 2 6
Note that when the polynomial is written as the sum of four terms, the product
of the first and last terms, (x2 26), is equal to the product of the two middle
terms, (22x 3x). We can apply these observations to factoring a trinomial into
two binomials.
EXAMPLE 3
Factor x2 1 7x 1 12.
Solution METHOD 1
(1) Write the trinomial as the sum of
four terms by writing 7x as the sum
of two terms whose product is
equal to the product of the first and
last terms:
(2) Rewrite the polynomial as the sum
of four terms:
x2 12 5 12x2
x 12x 5 12x2 but x 1 12x 7x ✘
2x 6x 5 12x2 but 2x 1 6x 7x ✘
3x 4x 5 12x2 and 3x 1 4x 5 7x ✔
x2 1
7x
1 12
2
5 x 1 3x 1 4x 1 12
(3) Factor out the common monomial
from the first two terms and from
the last two terms:
5 x(x 1 3) 1 4(x 1 3)
(4) Factor out the common binomial
factor:
5 (x 1 3)(x 1 4)
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METHOD 2
This trinomial can also be factored by recalling how the product of two binomials is found.
(1) The first term of the trinomial is the product of the first terms of the binomial factors:
x2 1 7x 1 12 5 (x
)(x
)
(2) The last term of the trinomial is the product of the last terms of the binomial factors. Write all possible pairs of factors for which this is true.
(x 1 1)(x 1 12)
(x 2 1)(x 2 12)
(x 1 2)(x 1 6)
(x 2 2)(x 2 6)
(x 1 3)(x 1 4)
(x 2 3)( x 2 4)
(3) For each possible pair of factors, find the product of the outside terms plus
the product of the inside terms.
12x 1 1x 5 13x ✘
212x 1 (21x) 5 213x ✘
6x 1 2x 5 8x ✘
26x 1 (22x) 5 28x ✘
3x 1 4x 5 7x ✔
23x 1 (24x) 5 27x ✘
(4) The factors of the trinomial are the two binomials such that the product of
the outside terms plus the product of the inside terms equals +7x.
x2 1 7x 1 12 5 (x 1 3)(x 1 4)
Answer (x 1 3)(x 1 4)
EXAMPLE 4
Factor: 3x2 2 x 2 4
Solution (1) Find the product of the first and
last terms:
(2) Find the factors of this product
whose sum is the middle term:
(3) Write the trinomial with four terms,
using this pair of terms in place of 2x:
Answer
3x2(24) 5 212x2
24x 1 3x 5 2x
3x2 2 x 2 4
5 3x2 2 4x 1 3x 2 4
(4) Factor the common factor from the
first two terms and from the last
two terms:
5 x(3x 2 4) 1 1(3x 2 4)
(5) Factor the common binomial factor:
5 (3x 2 4)(x 1 1)
(3x 2 4)(x 1 1)
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Factoring Polynomials
25
Special Products and Factors
We know that to multiply a binomial by a binomial, we perform four multiplications. If all four terms are unlike terms, then the polynomial is in simplest
form. Often, two of the four terms are similar terms that can be combined so
that the product is a trinomial. For example:
(a2 1 3)(a 2 2)
(x 1 3)(x 2 5)
2
5 a (a 2 2) 1 3(a 2 2)
3
5 x(x 2 5) 1 3(x 2 5)
2
5 x2 2 5x 1 3x 2 15
5 a 2 2a 1 3a 2 6
5 x2 2 2x 2 15
When the middle terms are additive inverses whose sum is 0, then the product of the two binomials is a binomial.
(a 1 3)(a 2 3) 5 a(a 2 3) 1 3(a 2 3)
5 a2 2 3a 1 3a 2 9
5 a2 1 0a 2 9
5 a2 2 9
Therefore, the product of the sum and difference of the same two numbers is the
difference of their squares. In general, the factors of the difference of two perfect squares are:
a2 2 b2 5 (a 1 b)(a 2 b)
EXAMPLE 5
Factor:
Think
Write
(2x) 2 2 (5) 2
a. 4x2 2 25
2
2
2
(2x 1 5)(2x 2 5)
b. 16 2 9y
(4) 2 (3y)
(4 1 3y)(4 2 3y)
c. 36a4 2 b4
(6a2) 2 2 (b2) 2
(6a2 1 b2)(6a2 2 b2)
When factoring a polynomial, it is important to make sure that each factor
is a prime polynomial or has no factors other than 1 and itself. Once you have
done this, the polynomial is said to be completely factored. For instance:
3ab2 2 6ab 1 3a
x4 2 16
5 3a(b2 2 2b 1 1)
5 (x2 1 4)(x2 2 4)
5 3a(b 2 1)(b 2 1)
5 (x2 1 4)(x 1 2)(x 2 2)
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EXAMPLE 6
Factor: 5a3 2 a2 2 5a 1 1
Solution (1) The product of the first and last terms
is equal to the product of the two
middle terms. Therefore, the polynomial
is a product of two binomials:
(2) Find a common factor of the first two
terms and then of the last two terms.
Then, factor out the common binomial
factor:
(3) The binomial factor (a2 2 1) is the
difference of two squares, which can be
factored into the sum and difference of
the equal factors of the squares:
?
5a3 ? 1 5 2a2 ? 25a
5a3 5 5a3 ✔
5a3 2 a2 2 5a 1 1
5 a2(5a 2 1) 2 1(5a 2 1)
5 (5a 2 1)(a2 2 1)
5 (5a 2 1)(a2 2 1)
5 (5a 2 1)(a 1 1)(a 2 1)
Answer (5a 2 1)(a 1 1)(a 2 1)
Exercises
Writing About Mathematics
1. Joel said that the factors of x2 1 bx 1 c are (x 1 d)(x 1 e) if de 5 c and d 1 e 5 b. Do you
agree with Joel? Justify your answer.
2. Marietta factored x2 1 5x 2 4 as (x 1 4)(x 1 1) because 4(1) 5 4 and 4 1 1 5 5. Do you
agree with Marietta? Explain why or why not.
Developing Skills
In 3–8, write each polynomial as the product of its greatest common monomial factor and a
polynomial.
3. 8x2 1 12x
4. 6a4 2 3a3 1 9a2
5. 5ab2 2 15ab 1 20a2b
6. x3y3 2 2x3y2 1 x2y2
7. 4a 2 12ab 1 16a2
8. 21a2 2 14a 1 7
In 9–26, write each expression as the product of two binomials.
9. y(y 1 1) 2 1(y 1 1)
10. 3b(b 2 2) 2 4(b 2 2)
11. 2x(y 1 4) 1 3(y 1 4)
12. a3 2 3a2 1 3a 2 9
13. 2x3 2 3x2 2 4x 1 6
14. y3 1 y2 2 5y 2 5
15. x2 1 7x 1 x 1 7
16. x2 1 5x 1 6
17. x2 2 5x 1 6
18. x2 1 5x 2 6
19. x2 2 x 2 6
20. x2 1 9x 1 20
21. 3x2 2 5x 2 12
22. 2y2 1 5y 2 3
23. 5b2 1 6b 1 1
24. 6x2 2 13x 1 2
25. 4y2 1 4y 1 1
26. 9x2 2 12x 1 4
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Quadratic Equations with Integral Roots
27
In 27–39, factor each polynomial completely.
27. a3 1 3a2 2 a 2 3
28. 5x2 2 15x 1 10
29. b3 2 4b
30. 4ax2 1 4ax 2 24a
31. 12c2 2 3
32. x4 2 81
33. x4 2 16
34. 2x3 1 13x2 1 15x
35. 4x3 2 10x2 1 6x
36. z4 2 12z2 1 27
37. (c 1 2) 2 2 1
38. 4 2 (y 2 1) 2
39. x2y 2 16y
40. 3(x 2 1)2 2 12
41. 9 2 9(x 1 2)2
Applying Skills
In 42–45, each polynomial represents the area of a rectangle. Write two binomials that could represent the length and width of the rectangle.
42. 4x2 2 7x 2 2
43. 16x2 2 25
44. 9x2 2 6x 1 1
45. 3x2 1 5x 2 2
1-7 QUADRATIC EQUATIONS WITH INTEGRAL ROOTS
An equation such as 3x 1 4 5 16 is a linear equation in one variable, that is, an
equation in which the variable occurs to the first power only. An equation such
as x2 2 3x 1 2 5 0 is a quadratic equation or a polynomial equation of degree
two because the highest power of the variable is two. A quadratic equation is in
standard form when it is written as a polynomial equal to 0. In general, if a 0,
the standard form of a quadratic equation is
ax2 1 bx + c 5 0
To write the quadratic equation 3 1 2x(x 2 1) 5 5 in standard form, first simplify the left member and then add 25 to each side of the equation.
3 1 x(x 2 1) 5 5
3 1 x2 2 x 5 5
3 1 x2 2 x 1 (25) 5 5 1 (25)
x2 2 x 2 2 5 0
Solving a Quadratic Equation
We know that ab 5 0 if and only if a 5 0 or b 5 0. We can use this fact to solve
a quadratic equation in standard form when the roots are integers. First, write
the non-zero member of the equation as the product of factors, each of which
contains the first power of the variable, and then set each factor equal to 0 to
find the roots.
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EXAMPLE 1
Solve the equation 3 1 x(x 2 1) 5 5.
Solution
3 1 x(x 2 1) 5 5
Write the equation in standard form.
x2 2 x 2 2 5 0
Factor the left side.
(x 2 2)(x 1 1) 5 0
x2250
x52
x1150
Set each factor equal to 0 and solve for x.
x 5 21
Check: x 5 2
3 1 x(x 2 1) 5 5
Check: x 5 21
3 1 x(x 2 1) 5 5
?
3 1 (21)(21 2 1) 5 5
?
3 2 1(22) 5 5
?
31255
3 1 2(2 2 1) 5 5
3 1 2(1) 5 5
31255
555✔
?
?
?
555✔
Answer x 5 2 or 21
EXAMPLE 2
Solve for x: 2x2 1 4x 5 30
2x2 1 4x 5 30
Solution (1) Write the equation in standard form:
2x2 1 4x 2 30 5 0
(2) Factor the left member:
2(x2 1 2x 2 15) 5 0
2(x 1 5)(x 2 3) 5 0
(3) Set each factor that contains the
variable equal to zero and solve
for x:
x1550
x 5 25
x2350
x53
Answer x 5 25 or 3
EXAMPLE 3
The length of a rectangle is 2 feet shorter than twice the width. The area of the
rectangle is 84 square feet. Find the dimensions of the rectangle.
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Quadratic Equations with Integral Roots
Solution
29
Let w 5 the width of the rectangle.
2w 2 2 5 the length of the rectangle.
Area 5 length 3 width
84 5 (2w 2 2)(w)
84 5 2w2 2 2w
0 5 2w2 2 2w 2 84
0 5 2(w2 2 w 2 42)
0 5 2(w 2 7)(w 1 6)
05w27
75w
05w16
–6 5 w
The width must be a positive number. Therefore, only 7 feet is a possible width
for the rectangle. When w 5 7, 2w 2 2 5 2(7) 2 2 5 12.
The area of the rectangle is 7(12) 5 84 square feet.
Answer The dimensions of the rectangle are 7 feet by 12 feet.
Exercises
Writing About Mathematics
1. Ross said that if (x 2 a)(x 2 b) 5 0 means that (x 2 a) 5 0 or (x 2 b) 5 0, then
(x 2 a)(x 2 b) 5 2 means that (x 2 a) 5 2 or (x 2 b) 5 2. Do you agree with Ross?
Explain why or why not.
2. If (x 2 a)(x 2 b)(x 2 c) 5 0, is it true that (x 2 a) 5 0, or (x 2 b) 5 0 or (x 2 c) 5 0?
Justify your answer.
Developing Skills
In 3–17, solve and check each of the equations.
3. x2 2 4x 1 3 5 0
4. x2 2 7x 1 10 5 0
5. x2 2 5x 2 6 5 0
6. x2 1 6x 1 5 5 0
7. x2 1 10x 2 24 5 0
8. x2 2 9x 5 10
9. 4 2 x(x 2 3) 5 0
10. x(x 1 7) 2 2 5 28
11. 2x2 2 x 5 12 1 x
12. 3x2 2 5x 5 36 2 2x
13. 7 5 x(8 2 x)
14. 9 5 x(6 2 x)
15. 2x(x 1 1) 5 12
16. x(x 2 2) 1 2 5 1
17. 3x(x 2 10) 1 80 5 5
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Applying Skills
18. Brad is 3 years older than Francis. The product of their ages is 154. Determine their ages.
19. The width of a rectangle is 12 feet less than the length. The area of the rectangle is 540
square feet. Find the dimensions of the rectangle.
20. The length of a rectangle is 6 feet less than three times the width. The area of the rectangle
is 144 square feet. Find the dimensions of the rectangle.
21. The length of the shorter leg, a, of a right triangle is 6 centimeters less than the length of the
hypotenuse, c, and the length of the longer leg, b, is 3 centimeters less than the length of the
hypotenuse. Find the length of the sides of the right triangle.
22. The height h, in feet, of a golf ball shot upward from a ground level sprint gun is described
by the formula h 5 216t2 1 48t where t is the time in seconds. When will the ball hit the
ground again?
1-8 QUADRATIC INEQUALITIES
DEFINITION
A quadratic inequality is an inequality that contains a polynomial of degree
two.
When we solve a linear inequality, we use the same procedure that we use to
solve a linear equation. Can we solve a quadratic inequality by using the same
procedure that we use to solve a quadratic equation? How is the solution of the
inequality x2 2 3x 2 4 . 0 similar to the solution of x2 2 3x 2 4 0?
To solve the equation, we factor the trinomial and write two equations in
which each factor is equal to 0. To solve the inequality, can we factor the trinomial and write two inequalities in which each factor is greater than 0?
x2 2 3x 2 4 5 0
x2 2 3x 2 4 0
(x 2 4)(x 1 1) 5 0
(x 2 4)(x 1 1) 0
?
x2450
x24 . 0
x1150
x11 . 0
?
If the product of two factors is greater than 0, that is, positive, then it is
true that each factor may be greater than 0 because the product of two
positive numbers is positive. However, it is also true that each factor may be
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Quadratic Inequalities
less than 0 because the product of two negative numbers is also positive.
Therefore, when we solve a quadratic inequality, we must consider two
possibilities:
x2 2 3x 2 4 0
(x 2 4)(x 1 1) 0
x 2 4 0 and x 1 1 0
x4
x 2 4 0 and x 1 1 0
x 21
x4
If x is greater than 4 and greater than
21, then x is greater than 4.
x 21
If x is less than 4 and less than 21,
then x is less than 21.
The solution set is {x : x 4 or x 21}.
On the number line, the solutions of the equality x2 2 3x 2 4 5 0 are 21
and 4. These two numbers separate the number line into three intervals.
4 3 2 1
0
1
2
3
4
5
6
7
8
Choose a representative number from each interval:
Let x 5 23:
Let x 5 1:
x2 2 3x 2 4 0
?
(23)2 2 3(23) 2 4 . 0
?
91924.0
14 0 ✔
Let x 5 6:
x2 2 3x 2 4 0
?
(1)2 2 3(1) 2 4 . 0
?
12324.0
26 0 ✘
x2 2 3x 2 4 0
?
(6)2 2 3(6) 2 4 . 0
?
36 2 18 2 4 . 0
14 0 ✔
We find that an element from the interval x 21 or an element from
the interval x 4 make the inequality true but an element from the interval
21 x 4 makes the inequality false.
A quadratic inequality in which the product of two linear factors is less than
zero is also solved by considering two cases. A product is negative if the two factors have opposite signs. Therefore, we must consider the case in which the first
factor is positive and the second factor is negative and the case in which the first
factor is negative and the second factor is positive.
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Procedure 1
To solve a quadratic inequality:
CASE 1
The quadratic inequality is of the form (x 2 a)(x 2 b) . 0
1. Let each factor be greater than 0 and solve the resulting inequalities.
2. Let each factor be less than 0 and solve the resulting inequalities.
3. Combine the solutions of the inequalities from steps 1 and 2 to find the
solution set of the given inequality.
CASE 2
The quadratic inequality is of the form (x 2 a)(x 2 b) , 0
1. Let the first factor be greater than 0 and let the second factor be less than
0. Solve the resulting inequalities.
2. Let the first factor be less than 0 and let the second factor be greater than
0. Solve the resulting inequalities.
3. Combine the solutions of the inequalities from steps 1 and 2 to find the
solution set of the given inequality.
A quadratic inequality can also be solved by finding the solutions to the corresponding equality. The solution to the inequality can be found by testing an
element from each interval into which the number line is separated by the roots
of the equality.
Procedure 2
To solve a quadratic inequality:
1. Find the roots of the corresponding equality.
2. The roots of the equality separate the number line into two or more
intervals.
3. Test a number from each interval. An interval is part of the solution if the
test number makes the inequality true.
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Quadratic Inequalities
EXAMPLE 1
List the solution set of x2 2 2x 2 15 0 if x is an element of the set of integers.
Solution Factor the trinomial. One of the factors is negative and other is positive.
x2 2 2x 2 15 0
(x 2 5)(x 1 3) 0
x 2 5 0 and x 1 3 0
x5
x 2 5 0 and x 1 3 0
x 23
x5
x23
The solution set is {x : 23 x 5}.
There are no values of x that are
both greater than 5 and less
than 23.
Check The numbers 23 and 5 separate the number line into three intervals. Choose a
representative number from each interval.
4 3 2 1
0
1
Let x 5 24:
2
3
4
5
6
7
8
Let x 5 1:
x 2 2x 2 15 0
2
?
(24)2 2 2(24) 2 15 , 0
?
16 1 8 2 15 , 0
9
0✘
Let x 5 7:
x 2 2x 2 15 0
2
?
(1)2 2 2(1) 2 15 , 0
?
1 2 2 2 15 , 0
216 0 ✔
x 2 2x 2 15 0
2
?
(7)2 2 2(7) 2 15 , 0
?
49 2 14 2 15 , 0
20 0 ✘
When we choose a representative number from each of these intervals, we find
that an element from the interval x 23 and an element from the interval
x 5 make the inequality false but an element from the interval 23 x 5
makes the inequality true.
Answer {–2, 21, 0, 1, 2, 3, 4}
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A graphing calculator can be used to
verify the quadratic inequality of the
examples. For instance, enter the inequality
x2 2 2x 2 15 0 into Y1. Use the 2nd
Y 1 = X 2- 2 X - 1 5 < 0
*
TEST menu to enter the inequality symbols
, , , and . Using the TRACE button, we
X=4
Y=1
can then verify that the integers from 22 to 4
make the inequality true (the “Y 5 1” in the
bottom of the graph indicates that the inequality is true) while integers less than
22 or greater than 4 make the inequality false (the “Y 5 0” in the bottom of the
graph indicates that the inequality is false).
EXAMPLE 2
List the solution set of x2 1 6x 1 8 $ 0 if x is an element of the set of integers.
Solution Factor the corresponding quadratic equality, x2 1 6x 1 8 5 0:
x2 1 6x 1 8 5 0
(x 1 2)(x 1 4) 5 0
x1250
x1450
x 5 22
x 5 24
The roots 22 and 24 separate the number line into three intervals: x 24,
24 x 22, and 22 x. Test a number from each interval to find the solution of the inequality:
Let x 5 25:
Let x 5 23:
?
x2 1 6x 1 8 $ 0
?
(25) 2 1 6(25) 1 8 $ 0
Let x 5 0:
?
x2 1 6x 1 8 $ 0
?
(23) 2 1 6(23) 1 8 $ 0
3$0✔
21 0 ✘
?
x2 1 6x 1 8 $ 0
?
(0) 2 1 6(0) 1 8 $ 0
8$0✔
The inequality is true in the intervals x 24 and x 22. However, since the
inequality is less than or equal to, the roots also make the inequality true.
Therefore, the solution set is {x : x 24 or x 22}.
Answer {. . ., 26, 25, 24, 22, 21, 0, . . .}
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Chapter Summary
35
Exercises
Writing About Mathematics
1. Rita said that when the product of three linear factors is greater than zero, all of the factors
must be greater than zero or all of the factors must be less than zero. Do you agree with
Rita? Explain why or why not.
2. Shelley said that if (x 2 7)(x 2 5) 0, then (x 2 7) must be the negative factor and
(x 2 5) must be the positive factor.
a. Do you agree with Shelly? Explain why or why not.
b. When the product of two factors is negative, is it always possible to tell which is the
positive factor and which is the negative factor? Justify your answer.
Developing Skills
In 3–17, write the solution set of each inequality if x is an element of the set of integers.
3. x2 1 5x 1 6 0
4. x2 1 5x 2 6 0
5. x2 2 3x 1 2 0
6. x2 2 7x 1 10 0
7. x2 2 x 2 6 0
8. x2 2 8x 2 20 0
9. x2 1 x 2 12 0
10. x2 2 6x 1 5 0
11. x2 2 2x 0
12. x2 2 x 6
13. x2 2 4x 1 4 0
14. x2 2 4x 1 4 0
15. x2 1 x 2 2 , 0
16. 2x2 2 2x 2 24 # 0
17. 2x2 2 2x 2 24 . 0
Applying Skills
18. A rectangular floor can be covered completely with tiles that each measure one square foot.
The length of the floor is 1 foot longer than the width and the area is less than 56 square
feet. What are the possible dimensions of the floor?
19. A carton is completely filled with boxes that are 1 foot cubes. The length of the carton is 2
feet greater than the width and the height of the carton is 3 feet. If the carton holds at most
72 cubes, what are the possible dimensions of the carton?
CHAPTER SUMMARY
The set of natural numbers is the set {1, 2, 3, 4, 5, 6, . . . }. The set of whole
numbers is the union of the set of natural numbers and the number 0. The set of
integers is the union of set of whole numbers and their opposites.
The absolute value of a is symbolized by a. If a 0, then a 5 a 2 0 5 a.
If a 0, then a 5 0 2 a 5 2a.
The domain is the set of numbers that can replace the variable in an algebraic expression. A number from the domain that makes an equation or
inequality true is a solution or root of the equation or inequality.
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We use the following properties of equality to solve an equation:
• Addition Property of Equality: If equals are added to equals, the sums are
equal.
• Subtraction Property of Equality: If equals are subtracted from equals, the
differences are equal.
• Multiplication Property of Equality: If equals are multiplied by equals,
the products are equal.
• Division Property of Equality: If equals are divided by equals, the quotients are equal.
We use the following properties of inequality to solve inequalities:
• Addition and Subtraction Property of Inequality: If equals are added to or
subtracted from unequals, the sums or differences are unequal in the same
order.
• Multiplication and Division Property of Inequality: If unequals are multiplied or divided by positive equals, the products or quotients are unequal
in the same order. If unequals are multiplied or divided by negative
equals, the products or quotients are unequal in the opposite order.
A monomial is a constant, a variable, or the product of constants and variables. The factors of a monomial are the numbers and variables whose product
is the monomial. A polynomial is the sum of monomials. Each monomial is a
term of the polynomial.
An absolute value equation or inequality can be solved by using the following relationships:
• If x 5 k for any positive number k, then x 5 2k or x 5 k.
• If x k for any positive number k, then 2k x k.
• If x k for any positive number k, then x k or x 2k.
If a 0, the standard form of a quadratic equation is ax2 1 bx + c 5 0. A
quadratic equation that has integral roots can be solved by factoring the polynomial of the standard form of the equations and setting each factor that contains the variable equal to zero.
An inequality of the form (x 2 a)(x 2 b) 0 can be solved by letting each
factor be positive and by letting each factor be negative. An inequality of the
form (x 2 a)(x 2 b) 0 can be solved by letting one factor be positive and the
other be negative.
VOCABULARY
1-1 Natural numbers • Counting numbers • Whole numbers • Opposite •
Additive inverse • Integers • Commutative group • Absolute value
1-2 Domain • Solution • Root • Equivalent equations
1-3 Monomial • Polynomial • Term • Similar terms • Like terms • Binomial •
Trinomial
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Review Exercises
37
1-4 Absolute value equation • Absolute value inequality
1-5 Base • Exponent • Power • FOIL
1-6 Factor • Common monomial factor • Common binomial factor • Prime
polynomials • Completely factored
1-7 Quadratic equation • Polynomial equation of degree two • Standard form
1-8 Quadratic Inequality
REVIEW EXERCISES
In 1–12, write each expression is simplest form.
1. 5x 2 7x
2. 4(2a 1 3) 2 9a
3. 2d 2 (5d 2 7)
4. 5(b 1 9) 2 3b(10 2 b)
5. x(x 1 3) 2 4(5 2 x)
6. 8 2 2(a2 1 a 1 4)
7. 7d(2d 1 c) 1 3c(4d 2 c)
8. (2x 2 1)(3x 1 1) 2 5x2
9. c2 2 (c 1 2)(c 2 2)
11. (22x)2 2 2x2
10. (2x 1 1)2 2 (2x 1 1)2
12. 4y2 1 2y(3y 2 2) 2 (3y)2
In 13–24, factor each polynomial completely.
13. 2x2 1 8x 1 6
14. 3a2 2 30a 1 75
15. 5x3 2 15x2 2 20x
16. 10ab2 2 40a
17. c4 2 16
18. 3y3 2 12y2 1 6y 2 24
19. x3 1 5x2 2 x 2 5
20. x4 2 2x2 2 1
21. 2x2 2 18x 1 36
22. x3 2 3x2 1 2x
23. 5a4 2 5b4
24. 5x2 1 22x 2 15
In 25–40, solve each equation or inequality for x. For each inequality, the solution set is a subset of the set of integers.
25. 8x 1 27 5 5x
26. 3(x 2 7) 5 5 1 x
27. 2x 2 9 5x 2 21
28. 23 2x 2 1 7
29. 2x 1 5 5 9
30. 7 2 x 1 1 5 0
31. 3 2 6y 1 2 11
32. 4 2 x 1 3 2
33. x2 2 9x 1 20 5 0
34. x(12 2 x) 5 35
35. x2 1 7x 1 6 0
36. x2 2 2x 2 35 0
37. x2 5x
38. x(x 1 3) 0
39. 4x2 2 16x 1 12 # 0
40. 2x2 1 2x 2 4 $ 0
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The Integers
41. Explain why the equation 3x 2 5 1 4 5 0 has no solution in the set of
integers.
42. The length of a rectangle is 4 centimeters less than three times the width.
The perimeter of the rectangle is 88 centimeters. What are the dimensions
of the rectangle?
43. The length of a rectangle is 6 feet more than three times the width. The
area of the rectangle is 240 square feet. What are the dimensions of the
rectangle?
44. The length of the longer leg of a right triangle is 4 inches more than twice
the length of the shorter leg. The length of the hypotenuse is 6 inches
more than twice the length of the shorter leg. What are the lengths of the
legs of the right triangle?
45. The equation h 5 216t2 1 80t gives the height, h, in feet after t seconds
when a ball has been thrown upward at a velocity of 80 feet per second.
a. Find the height of the ball after 3 seconds.
b. After how many seconds will the ball be at a height of 64 feet?
Exploration
A whole number that is the sum of all of its factors except itself is called a
perfect number. Euclid said that if (2k 2 1) is a prime, then N 5 2k21(2k 2 1) is
a perfect number. A perfect number of this form is called a Euclidean perfect
number.
1. Use the formula for a Euclidean perfect number to find the first four perfect numbers.
2. Show that a Euclidean perfect number is always even.
3. Show that a Euclidean perfect number must have 6 or 8 as the units digit.
(Hint: What are the possible units digits of (2k 2 1)? Of 2k21?)
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CHAPTER
THE
RATIONAL
NUMBERS
When a divided by b is not an integer, the quotient
is a fraction.The Babylonians, who used a number system based on 60, expressed the quotients:
1
20 8 as 2 1 30
60 instead of 22
30
5
21 8 as 2 1 37
60 1 3,600 instead of 28
Note that this is similar to saying that 20 hours
divided by 8 is 2 hours, 30 minutes and that 21 hours
divided by 5 is 2 hours, 37 minutes, 30 seconds.
This notation was also used by Leonardo of Pisa
(1175–1250), also known as Fibonacci.
The base-ten number system used throughout the
world today comes from both Hindu and Arabic mathematicians. One of the earliest applications of the
base-ten system to fractions was given by Simon Stevin
(1548–1620), who introduced to 16th-century Europe
a method of writing decimal fractions. The decimal
that we write as 3.147 was written by Stevin as
3 1 4 7 or as 3 1 4 7 . John Napier
(1550–1617) later brought the decimal point into common usage.
2
CHAPTER
TABLE OF CONTENTS
2-1 Rational Numbers
2-2 Simplifying Rational
Expressions
2-3 Multiplying and Dividing
Rational Expressions
2-4 Adding and Subtracting
Rational Expressions
2-5 Ratio and Proportion
2-6 Complex Rational
Expressions
2-7 Solving Rational Equations
2-8 Solving Rational Inequalities
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
39
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2-1 RATIONAL NUMBERS
When persons travel to another country, one of the first things that they learn is
the monetary system. In the United States, the dollar is the basic unit, but
most purchases require the use of a fractional part of a dollar. We know
5
1
1
5 20
that a penny is $0.01 or 100
of a dollar, that a nickel is $0.05 or 100
of a
10
1
5 10
dollar, and a dime is $0.10 or 100
of a dollar. Fractions are common in our
everyday life as a part of a dollar when we make a purchase, as a part of a pound
when we purchase a cut of meat, or as a part of a cup of flour when we are
baking.
In our study of mathematics, we have worked with numbers that are not
8
2
1
integers. For example, 15 minutes is 15
60 or 4 of an hour, 8 inches is 12 or 3 of a foot,
8
1
and 8 ounces is 16 or 2 of a pound. These fractions are numbers in the set of
rational numbers.
DEFINITION
A rational number is a number of the form ba where a and b are integers and
b 0.
For every rational number ba that is not equal to zero, there is a multiplicative
inverse or reciprocal ba such that ba ? ba 5 1. Note that ba ? ba 5 ab
ab. If the non-zero
numerator of a fraction is equal to the denominator, then the fraction is equal
to 1.
EXAMPLE 1
Write the multiplicative inverse of each of the following rational numbers:
Answers
a. 34
b. 25
8
c. 5
4
3
8
25
1
5
5 285
Note that in b, the reciprocal of a negative number is a negative number.
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Rational Numbers
Decimal Values of Rational Numbers
The rational number ba is equivalent to a b. When a fraction or a division such
as 25 100 is entered into a calculator, the decimal value is displayed. To
express the quotient as a fraction, select Frac from the MATH menu. This can be
done in two ways.
ENTER: 25
MATH
DISPLAY:
ENTER: 25
100 ENTER
1
100 MATH
ENTER
ENTER
DISPLAY:
25/100
25/100
.25
Ans
1
Frac
1/4
Frac
1/4
8
When a calculator is used to evaluate a fraction such as 12
or 8 12, the decimal value is shown as .6666666667. The calculator has rounded the value to ten
8
decimal places, the nearest ten-billionth. The true value of 12
, or 23, is an infinitely
repeating decimal that can be written as 0.6. The line over the 6 means that the
digit 6 repeats infinitely. Other examples of infinitely repeating decimals are:
0.2 5 0.22222222 . . . 5 29
0.142857 5 0.142857142857 . . . 5 17
4
0.12 5 0.1212121212 . . . 5 33
0.16 5 0.1666666666 . . . 5 16
Every rational number is either a finite decimal or an infinitely repeating
decimal. Because a finite decimal such as 0.25 can be thought of as having an
infinitely repeating 0 and can be written as 0.250, the following statement is true:
A number is a rational number if and only if it can be written as an infinitely
repeating decimal.
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EXAMPLE 2
Find the common fractional equivalent of 0.18.
Solution Let x 5 0.18 5 0.18181818 . . .
How to Proceed
(1) Multiply the value of x by 100 to write a
number in which the decimal point follows
the first pair of repeating digits:
(2) Subtract the value of x from both sides of
this equation:
(3) Solve the resulting equation for x and simplify
the fraction:
Check The solution can be checked on a calculator.
ENTER: 2
DISPLAY:
100x 5 18.181818 . . .
100x 5 18.181818c
2x 5 20.181818c
99x 5 18
2
x 5 18
99 5 11
11 ENTER
2/11
.1818181818
Answer
2
11
EXAMPLE 3
Express 0.1248 as a common fraction.
Solution: Let x 5 0.12484848. . .
How to Proceed
(1) Multiply the value of x by the power of 10 that
makes the decimal point follow the first set of
repeating digits. Since we want to move the
decimal point 4 places, multiply by 104 5 10,000:
(2) Multiply the value of x by the power of 10 that
makes the decimal point follow the digits that
do not repeat. Since we want to move the
decimal point 2 places, multiply by 102 5 100:
(3) Subtract the equation in step 2 from the
equation in step 1:
(4) Solve for x and reduce the fraction to
lowest terms:
Answer
103
825
10,000x 5 1,248.4848. . .
100x 5 12.4848. . .
10,000x 5 1,248.4848c
2100x 5 212.4848c
9,900x 5 1,236
1,236
x 5 9,900 5 103
825
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Rational Numbers
43
Procedure
To convert an infinitely repeating decimal to a common fraction:
1. Write the equation: x 5 decimal value.
2. Multiply both sides of the equation in step 1 by 10m, where m is the number
of places to the right of the decimal point following the first set of repeating
digits.
3. Multiply both sides of the equation in step 1 by 10n, where n is the number
of places to the right of the decimal point following the non-repeating digits.
(If there are no non-repeating digits, then let n 5 0.)
4. Subtract the equation in step 3 from the equation in step 2.
5. Solve the resulting equation for x, and simplify the fraction completely.
Exercises
Writing About Mathematics
1. a. Why is a coin that is worth 25 cents called a quarter?
b. Why is the number of minutes in a quarter of an hour different from the number of
cents in a quarter of a dollar?
2. Explain the difference between the additive inverse and the multiplicative inverse.
Developing Skills
In 3–7, write the reciprocal (multiplicative inverse) of each given number.
3. 83
7
4. 12
5. 22
7
6. 8
7. 1
In 8–12, write each rational number as a repeating decimal.
8. 61
9. 29
10. 75
2
11. 15
12. 78
In 13–22, write each decimal as a common fraction.
13. 0.125
14. 0.6
15. 0.2
16. 0.36
17. 0.108
18. 0.156
19. 0.83
20. 0.57
21. 0.136
22. 0.1590
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2-2 SIMPLIFYING RATIONAL EXPRESSIONS
A rational number is the quotient of two integers. A rational expression is the
quotient of two polynomials. Each of the following fractions is a rational expression:
3
4
ab
7
x 1 5
2x
y 2 2
2
y 2 5y 1 6
a2 2 1
4ab
Division by 0 is not defined. Therefore, each of these rational expressions
has no meaning when the denominator is zero. For instance:
1 5
• x 2x
has no meaning when x 5 0.
2 2 1
• a 4ab
has no meaning when a 5 0 or when b 5 0.
y 2 2
y 2 2
• y2 2 5y 1 6 5 (y 2 2)(y 2 3) has no meaning when y 5 2 or when y 5 3.
EXAMPLE 1
2 5
For what value or values of a is the fraction 3a2 2a
2 4a 1 1 undefined?
Solution A fraction is undefined or has no meaning when a factor of the denominator
is equal to 0.
How to Proceed
3a2 2 4a 1 1 5 (3a 2 1)(a 2 1)
(1) Factor the denominator:
(2) Set each factor equal to 0:
3a 2 1 5 0
a2150
(3) Solve each equation for a:
3a 5 1
a51
a5
1
3
Answer The fraction is undefined when a 5 13 and when a 5 1.
If ba and dc are rational numbers with b 0 and d 0, then
a
b
ac
? dc 5 bd
and
ac
bd
5 ab ? dc 5 ad ? bc
3
3
1
3
1
1
1
5 33 3
For example: 34 3 13 5 12
and 12
3 4 5 3 3 4 5 1 3 4 5 4.
We can write a rational expression in simplest form by finding common factors in the numerator and denominators, as shown above.
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Simplifying Rational Expressions
45
EXAMPLE 2
Simplify:
Answers
a. 2x
6
5 22 ? x3 5 1 ? x3 5 x3
b. a(a3ab
2 1)
3b
3b
5 aa ? a 3b
2 1 5 1 ? a 2 1 5 a 2 1 (a 2 0, 1)
y 2 2
c. y2 2 5y 1 6
y 2 2
1
y 2 2
1
1
5 (y 2 2)(y 2 3) 5 y 2 2 ? y 2 3 5 1 ? y 2 3 5 y 2 3 (y 2 2, 3)
Note: We must eliminate any value of the variable or variables for which the
denominator of the given rational expression is zero.
1
The rational expressions x3, a 3b
2 1, and y 2 3 in the example shown above are
in simplest form because there is no factor of the numerator that is also a factor
of the denominator except 1 and 21. We say that the fractions have been
reduced to lowest terms.
When the numerator or denominator of a rational expression is a monomial, each number or variable is a factor of the monomial. When the numerator
or denominator of a rational expression is a polynomial with more than one
term, we must factor the polynomial. Once both the numerator and denominator of the fraction are factored, we can reduce the fraction by identifying factors
in the numerator that are also factors in the denominator.
In the example given above, we wrote:
y 2 2
y2 2 5y 1 6
y 2 2
y 2 2
1
1
1
5 (y 2 2)(y 2 3) 5 y 2 2 ? y 2 3 5 1 ? y 2 3 5 y 2 3 (y 2 2, 3)
We can simplify this process by canceling the common factor in the numerator and denominator.
1
(y 2 2)
(y 2 2) (y 2 3)
1
5y2
3 (y 2, 3)
1
Note that canceling (y 2 2) in the numerator and denominator of the fraction given above is the equivalent of dividing (y 2 2) by (y 2 2). When any number or algebraic expression that is not equal to 0 is divided by itself, the quotient
is 1.
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Procedure
To reduce a fraction to lowest terms:
METHOD 1
1. Factor completely both the numerator and the denominator.
2. Determine the greatest common factor of the numerator and the
denominator.
3. Express the given fraction as the product of two fractions, one of which has
as its numerator and its denominator the greatest common factor determined in step 2.
4. Write the fraction whose numerator and denominator are the greatest
common factor as 1 and use the multiplication property of 1.
METHOD 2
1. Factor both the numerator and the denominator.
2. Divide both the numerator and the denominator by their greatest common
factor by canceling the common factor.
EXAMPLE 3
2 12
Write 3x 3x
in lowest terms.
Solution
METHOD 1
METHOD 2
1
3x 2 12
3x
5
3(x 2 4)
3x
5
3
3
3x 2 12
3x
5
3 (x 2 4)
3x
5
x 2 4
x
1
5
5
Answer
x 2 4
x
4
?x2
x
4
1?x2
x
x 2 4
x
(x 0)
Factors That Are Opposites
The binomials (a 2 2) and (2 2 a) are opposites or additive inverses. If we
change the order of the terms in the binomial (2 2 a), we can write:
(2 2 a) 5 (2a 1 2) 5 21(a 2 2)
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Simplifying Rational Expressions
47
We can use this factored form of (2 2 a) to reduce the rational expression
to lowest terms.
2 2 a
a2 2 4
1
2 2 a
a2 2 4
21(a 2 2)
21(a 2 2)
1
21
5 (a 1 2)(a 2 2) 5 (a 1 2)(a 2 2) 5 (a 1 2) 5 2a 1 2 (a 2 2, 22)
1
EXAMPLE 4
2x
Simplify the expression: x2 28 2
8x 1 16
Solution
METHOD 1
8 2 2x
x2 2 8x 1 16
5
METHOD 2
2(4 2 x)
(x 2 4)(x 2 4)
8 2 2x
x2 2 8x 1 16
2(21)(x 2 4)
5 (x 2 4)(x 2 4)
(x 2 4)
2(4 2 x)
5 (x 2 4)(x 2 4)
2(21)(x 2 4)
5 (x 2 4)(x 2 4)
1
22
5 (x 2 4) ? (x 2 4)
22(x 2 4)
5 (x 2 4)(x 2 4)
1
5
Answer
22
x 2 4
22
x 2 4
5
22
x 2 4
(x 4)
Exercises
Writing About Mathematics
1
1. Abby said that 3x3x
1 4 can be reduced to lowest terms by canceling 3x so that the result is 4.
Do you agree with Abby? Explain why or why not.
2 3
2. Does 2a
2a 2 3 5 1 for all real values of a? Justify your answer.
Developing Skills
In 3–10, list the values of the variables for which the rational expression is undefined.
2
3. 5a
3a
4. 22d
6c
1 2
5. a ab
7. 2a 2a
2 7
8. b2 b1 1b 32 6
9. 2c2 4c
2 2c
2 5
6. xx 1
5
10. x3 2 5x52 2 6x
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The Rational Numbers
In 11–30, write each rational expression in simplest form and list the values of the variables for
which the fraction is undefined.
12xy2
4
14. 14b
21b3
9y2 1 3y
6y2
2 4b2
18. 8ab6ab
6
11. 10
2b
12. 5a
10a
13. 3x2y
9cd2
15. 12c
4 2
d
1 16
16. 8a 12a
17.
19. 15d 10d
2 20d2
20. 8c2 8c
1 16c
21. 9xy 1 6x2y3
2
16
23. 4a4a 2
1 8
2
7x 1 12
24. xx2 2
1 2x 2 15
25. y2 1 4y 1 4
3
2 2 a 1 1
27. a a22 2a 2a
1 1
28.
3xy
2
2a 1 10
22. 3a
1 15
5y2 2 20
3 2 (b 1 1)
4 2 b2
27 1 7a
26. 21a
2
2 21
4 2 2(x 2 1)
29. x2 2 6x 1 9
30.
5(1 2 b) 1 15
b2 2 16
2-3 MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS
Multiplying Rational Expressions
8
We know that 23 3 45 5 15
and that 34 3 32 5 98. In general, the product of two
ac
rational numbers ba and dc is ba ? dc 5 bd
for b 0 and d 0.
This same rule holds for the product of two rational expressions:
The product of two rational expressions is a fraction whose numerator is the
product of the given numerators and whose denominator is the product of
the given denominators.
For example:
3(a 1 5)
4a
12
36(a 1 5)
? a 5
(a 0)
4a2
This product can be reduced to lowest terms by dividing numerator and denominator by the common factor, 4.
9
3(a 1 5)
4a
36(a 1 5)
36 (a 1 5)
9(a 1 5)
? 12
5
5
a 5
4a2
4a2
a2
1
We could have canceled the factor 4 before we multiplied, as shown below.
3(a 1 5)
4a
12
3 a 5
3(a 1 5)
4a
1
3
9(a 1 5)
3 12
a 5
a2
Note that a is not a common factor of the numerator and denominator because
it is one term of the factor (a 1 5), not a factor of the numerator.
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49
Procedure
To multiply fractions:
METHOD 1
1. Multiply the numerators of the given fractions and the denominators of the
given fractions.
2. Reduce the resulting fraction, if possible, to lowest terms.
METHOD 2
1. Factor any polynomial that is not a monomial.
2. Cancel any factors that are common to a numerator and a
denominator.
3. Multiply the resulting numerators and the resulting denominators to write
the product in lowest terms.
EXAMPLE 1
2
b2 2 9
a. Find the product of 5b12b
in simplest form.
2
1 15 ?
3b 1 9b
b. For what values of the variable are the given fractions and the product
undefined?
Solution a. METHOD 1
How to Proceed
(1) Multiply the numerators of
the fractions and the
denominators of the
fractions:
(2) Factor the numerator and
the denominator. Note that
the factors of (5b 1 15) are
5(b 1 3) and the factors of
(3b2 1 9b) are 3b(b 1 3).
Reduce the resulting
fraction to lowest terms:
12b2
5b 1 15
b2 2 9
12b2 (b2 2 9)
? 3b2 1 9b 5 (5b 1 15)(3b2 1 9b)
12b2 (b 1 3)(b 2 3)
5 15b(b 1 3)(b 1 3)
3b(b 1 3)
4b(b 2 3)
5 3b(b 1 3) ? 5(b 1 3)
4b(b 2 3)
5 1 ? 5(b 1 3)
4b(b 2 3)
5 5(b 1 3) Answer
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The Rational Numbers
METHOD 2
How to Proceed
12b2
5b 1 15
(1) Factor each binomial term
completely:
2 2 9
12b2
(b 1 3)(b 2 3)
? 3bb2 1
3b(b 1 3)
9b 5 5(b 1 3) ?
(2) Cancel any factors that are
common to a numerator
and a denominator:
5
(3) Multiply the remaining
factors:
5 5(b 1 3) Answer
4 b
12b2
5(b 1 3)
1
?
(b 1 3) (b 2 3)
3b (b 1 3)
11
1
4b(b 2 3)
b. The given fractions and their product are undefined when b 5 0 and
when b 5 23. Answer
Dividing Rational Expressions
We can divide two rational numbers or two rational expressions by changing the
division to a related multiplication. Let ba and dc be two rational numbers or rational expressions with b 0, c 0, d 0. Since dc is a non-zero rational number or
rational expression, there exists a multiplicative inverse dc such that dc ? dc 5 1.
a?d
a?d
a
a d
c
a
b
b ? c 5 b c 5 b c 5 a ?d
4
5
5
1
c d
b c
c
c d
b
d
d?c
d
d c
We have just derived the following procedure.
Procedure
To divide two rational numbers or rational expressions, multiply the
dividend by the reciprocal of the divisor.
For example:
x 1 5
2x2
5
x 1 5 x 1 1
4x1
1 5 2x2 ?
5
5
(x 1 5)(x 1 1)
10x2
(x 21, 0)
This product is in simplest form because the numerator and denominator have
no common factors.
Recall that a can be written as a1 and therefore, if a 0, the reciprocal of a
1
is a. For example:
1
2b 2 2
5
4 (b 2 1) 5
2(b 2 1)
5
1
?b215
2(b 2 1)
5
1
2
?b2
1 5 5 (b 1)
1
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Multiplying and Dividing Rational Expressions
EXAMPLE 2
2
2
2 10
5a
Divide and simplify: a 2 3a
4a 2
5a
15
Solution
How to Proceed
(1) Use the reciprocal of
the divisor to write
the division as a
multiplication:
a2 2 3a 2 10
5a
2
2
5a
2 10
15
4a 2
5 a 2 3a
? a2 2
15
5a
5a
(2) Factor each polynomial:
Answer
5
(3) Cancel any factors that
are common to the
numerator and
denominator:
(a 2 5)(a 1 2)
5a
5
(a 2 5) (a 1 2)
5a
(4) Multiply the remaining
factors:
5
3(a 1 2)
a2
15
? a(a 2 5)
3
1
15
? a(a 2 5)
1
1
3(a 1 2)
2
a
(a 0, 5)
EXAMPLE 3
Perform the indicated operations and write the answer in simplest form:
3a
a 1 3
3
4 2a 5a
1 6 ? a
Solution Recall that multiplications and divisions are performed in the order in which
they occur from left to right.
3a
a 1 3
3
3a
4 2a 5a
1 6 ? a 5 a 1 3 ?
1
2(a 1 3)
5a
? a3
1
a
5 a 31 3 ?
1
2(a 1 3)
5a
? a3
1
2
5 65 ? a3
1
5
2a
5
(a 2 23, 0) Answer
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Exercises
Writing About Mathematics
4(x 2 2)
3
1. Joshua wanted to write this division in simplest form: x 2
. He began by cancel7
2 4
ing (x 2 2) in the numerator and denominator and wrote following:
1
3
x 2 2
4
4(x 2 2)
7
5 31 4 47 5 31 3 74 5 21
4
1
Is Joshua’s answer correct? Justify your answer.
12x 4 4
3x
4
2. Gabriel wrote 5x12x
1 10 4 5 5 (5x 1 10) 4 5 5 x 1 2. Is Gabriel’s solution correct? Justify
your answer.
Developing Skills
In 3–12, multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined.
3. 23 3 34
5
? 3a
4. 7a
20
x
5. 5x ? 8y
4y
10
6. 3a
5 ? 9a
1
? 5b 12
7. b 1
4
1 5
2
a2
8. a 23a100 ? 2a 2
20
9.
7y 1 21
7y
3
? y2 2 9
2
1 4
? a2 3a
11. 2a 6a
1 2a
2
5a 1 4
1 4
? a2a
10. a 2
2
3a 1 6
2 16
1 5x
12. 6x2222x9 ? 15 4x
In 13–24, divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined.
9
13. 34 4 20
6
14. 12
a 4 4a
3b
15. 6b
5c 4 10c
a2
4 3a
16. 8a
4
2 2
4 4x 92 8
17. x 3x
18.
2 2 6c 1 9
c 2 3
19. c 5c
2 15 4 5
2 2 w
2
4 w 52 1
20. w 5w
12
4 (b 1 3)
21. 4b 1
b
2
1 15
4 (a 1 3)
22. a 1 8a
4a
23. (2x 1 7) 4 2x2 1 15x 2 7
24. (a2 2 1) 4 2a a1 2
6y2 2 3y
3y
4
4y2 2 1
2
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53
In 25–30, perform the indicated operations and write the result in simplest form. In each case, list
any values of the variables for which the fractions are not defined.
25. 35 3 59 4 43
x2 2 1
1
4x1
26. x 3x
3
x
2 1 ?
a2 2 4
a 2 2
27. a 2a
1 2 ? 4a2 4 a
1 x 1 4
28. (x2 2 2x 1 1) 4 x 2
? 3x
3
2b 1 4
29. (3b) 2 4 b 3b
1 2 ?
b
2
2
3x 1 2
x 2 1
30. x 2 4x
? x212x
2 2x 4 x
2-4 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS
We know that 37 5 3 A 17 B and that 27 5 2 A 17 B . Therefore:
3
7
1 27 5 3 A 17 B 1 2 A 17 B
5 (3 1 2) A 17 B
5 5 A 17 B
5 57
We can also write:
x 1 2
x
1 2x x2 5 5 (x 1 2) A x1 B 1 (2x 2 5) A x1 B
5 (x 1 2 1 2x 2 5) A x1 B
5 3x x2 3 (x 2 0)
In general, the sum of any two fractions with a common denominator is a
fraction whose numerator is the sum of the numerators and whose denominator is the common denominator.
a
4a
1a1
1 5 a 1 1 (a 21)
In order to add two fractions that do not have a common denominator, we
need to change one or both of the fractions to equivalent fractions that do have
a common denominator. That common denominator can be the product of the
given denominators.
For example, to add 51 1 17, we change each fraction to an equivalent fraction
whose denominator is 35 by multiplying each fraction by a fraction equal to 1.
3a
a 1 1
1
5
1 17 5 15 3 77 1 17 3 55
7
5
5 35
1 35
5 12
35
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5
To add the fractions 2x
and y3 , we need to find a denominator that is a multiple of both 2x and of y. One possibility is their product, 2xy. Multiply each fraction by a fraction equal to 1 so that the denominator of each fraction will be 2xy:
y
5y
5
5
2x 5 2x ? y 5 2xy
5y
5
3
6x
2x 1 y 5 2xy 1 2xy
3
y
5y 1 6x
2xy
and
5
6x
5 y3 ? 2x
2x 5 2xy
(x 0, y 0)
The least common denominator (LCD) is often smaller than the product
of the two denominators. It is the least common multiple (LCM) of the
denominators, that is, the product of all factors of one or both of the
denominators.
1
For example, to add 2a 11 2 1 a2 2
1, first find the factors of each denominator. The least common denominator is the product of all of the factors of the first
denominator times all factors of the second that are not factors of the first. Then
multiply each fraction by a fraction equal to 1 so that the denominator of each
fraction will be equal to the LCD.
Factors of 2a 1 2: 2 (a 1 1)
(a 1 1) (a 2 1)
Factors of a2 2 1:
LCD: 2 (a 1 1) (a 2 1)
1
2a 1 2
1
5 2(a 11 1) ? aa 2
2 1
and
1
a2 2 1
2 1
5 2(a 1a 1)(a
2 1)
1
2
5 (a 1 1)(a
2 1) ? 2
2
5 2(a 1 1)(a
2 1)
a 2 1 1 2
a 1 1
1
1 a2 2
1 5 2(a 1 1)(a 2 1) 5 2(a 1 1)(a 2 1)
Since this sum has a common factor in the numerator and denominator, it
can be reduced to lowest terms.
1
2a 1 2
1
a 1 1
2(a 1 1)(a 2 1)
1 1
1
5 2(a 1a 1)(a
2 1) 5 2(a 2 1) (a 21, a 1)
1
Any polynomial can be written as a rational expression with a denominator
of 1. To add a polynomial to a rational expression, write the polynomial as an
equivalent rational expression.
1
For example, to write the sum b 1 3 1 2b
as a single fraction, multiply
2b
(b 1 3) by 1 in the form 2b.
3 2b
1
1
(b 1 3) 1 2b
5b1
1 A 2b B 1 2b
2 1 6b
1
5 2b 2b
1 2b
2
5 2b 12b6b 1 1 (b 2 0)
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EXAMPLE 1
x
x
Write the difference x2 2 4x
1 3 2 x2 1 2x 2 3 as a single fraction in lowest terms.
Solution
How to Proceed
(1) Find the LCD of the
fractions:
x2 2 4x 1 3 5 (x 2 3) (x 2 1)
(x 2 1) (x 1 3)
x2 1 2x 2 3 5
LCD 5 (x 2 3) (x 2 1) (x 1 3)
x
x2 2 4x 1 3
(2) Write each fraction as an
equivalent fraction with a
denominator equal to the
LCD:
x
x 1 3
5 (x 2 3)(x
2 1) ? x 1 3
2
x 1 3x
5 (x 2 3)(x
2 1)(x 1 3)
x 2 3
x2 1 2x 2 3
2 3
x 2 3
5 (x 1x3)(x
2 1) ? x 2 3
2
2 6x 1 9
5 (x 2 x3)(x
2 1)(x 1 3)
x
x2 2 4x 1 3
(3) Subtract:
3
2 x2 1x 2
2x 2 3
x2 1 3x 2 (x2 2 6x 1 9)
5 (x 2 3)(x 2 1)(x 1 3)
Answer
(4) Simplify:
9x 2 9
5 (x 2 3)(x
2 1)(x 1 3)
(5) Reduce to lowest terms:
9
5 (x 2 3)(x
1 3)
9
(x 2 3)(x 1 3)
(x 23, 1, 3)
EXAMPLE 2
1
Simplify: A x 2 x1 B A 1 1 x 2
1B
Solution STEP 1. Rewrite each expression in parentheses as a single fraction.
x 2 x1 5 x A xx B 2 x1
2
5 xx 2 x1
x 2 1
1
1
11x2
1 5 1Ax 2 1B 1 x 2 1
1
1
5 xx 2
2 1 1 x 2 1
1 1 1
5x2
x 2 1
and
2
5 x x2 1
x
5x2
1
STEP 2. Multiply.
Ax
2
x
2 1
x B Ax 2 1B
5Q
(x 1 1)(x 2 1)
x
R Ax 2
x
1B
5 ¢
1
1
x
(x 1 1)(x 2 1)
≤ ¢x 2 1≤
x
1
1
5x1
1
5x11
1
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Alternative STEP 1. Multiply using the distributive property.
Solution
A x 2 x1 B A 1 1 x 21 1 B 5 x A 1 1 x 21 1 B 2 x1 A 1 1
1
x 2 1B
x
1
1
5x1x2
1 2 x 2 x(x 2 1)
STEP 2. Add the fractions. The least common denominator is x(x 2 1).
x(x 2 1)
x
x
1
1
1
1
1
x 1 x 2 1 2 x 2 x(x 2 1) 5 xQ x(x 2 1) R 1 (x 2 1) A xx B 2 x A xx 2
2 1 B 2 x(x 2 1)
x3 2 x2
x2
x 2 1
1
5 x(x
2 1) 1 x(x 2 1) 2 x(x 2 1) 2 x(x 2 1)
3
2
x2 2 x 1 1 2 1
5 x 2 x 1x(x
2 1)
x3 2 x
5 x(x
2 1)
STEP 3. Simplify.
1
x3 2 x
x(x 2 1)
5
1
x (x 2 1)(x 1 1)
x (x 2 1)
1
5x11
1
Answer x 1 1 (x 0, 1)
Exercises
Writing About Mathematics
(a 1 2)(a 2 1)
2
1. Ashley said that (a 1 3)(a 2 1) 5 aa 1
1 3 for all values of a except a 5 23. Do you agree with
Ashley? Explain why or why not.
1 bc
2. Matthew said that ba 1 dc 5 ad bd
when b 0, d 0. Do you agree with Matthew? Justify
your answer.
Developing Skills
In 3–20, perform the indicated additions or subtractions and write the result in simplest form. In
each case, list any values of the variables for which the fractions are not defined.
3. x3 1 2x
3
2
2
4. 2x 5x1 1 2 7x 5x2 1
1
1
2a1
6. a 2
4
5
7.
1 3
2 2
9. 2x6x
2 x 4x
y 1 2
2
1
2y 2 3
3
5. x7 1 x3
1 5
2 8
2 a 8a
8. a 5a
1 1
10. a 3a
1 32
11. 3 1 x2
1
12. 5 2 2y
3
13. a 2 2a
14. x1 1 1
3
x 2 2
15. x 1
2 1 x
b
1
16. b 2
1 2 2 2 2b
2
1
17. 2 2
x 1 x 2 2
18. a2 2 1a 2 6 2 2a2 2 17a 1 3
2
1
19. a2 2
4 2 a2 1 2a
1
2
20. x1 1 x 2
2 2 x2 2 2x
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57
Applying Skills
In 21–24, the length and width of a rectangle are expressed in terms of a variable.
a. Express each perimeter in terms of the variable.
b. Express each area in terms of the variable.
21. l 5 2x and w 5 x1
22. l 5 3x 1 3 and w 5 31
3
x
23. l 5 x 2
1 and w 5 x 2 1
x
x
24. l 5 x 1
1 and w 5 x 1 2
2-5 RATIO AND PROPORTION
We often want to compare two quantities that use the same unit. For example,
in a given class of 25 students, there are 11 students who are boys. We can say
11
that 25
of the students are boys or that the ratio of students who are boys to all
students in the class is 11 : 25.
DEFINITION
A ratio is the comparison of two numbers by division. The ratio of a to b can
be written as ba or as a : b when b 0.
A ratio, like a fraction, can be simplified by dividing each term by the same
non-zero number. A ratio is in simplest form when the terms of the ratio are
integers that have no common factor other than 1.
For example, to write the ratio of 3 inches to 1 foot, we must first write each
measure in terms of the same unit and then divide each term of the ratio by a
common factor.
3 inches
1 foot
1 foot
3
1
5 31inches
foot 3 12 inches 5 12 5 4
In lowest terms, the ratio of 3 inches to 1 foot is 1 : 4.
An equivalent ratio can also be written by multiplying each term of the ratio
by the same non-zero number. For example, 4 : 7 5 4(2) : 7(2) 5 8 : 14.
In general, for x 0:
a : b 5 ax : bx
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EXAMPLE 1
The length of a rectangle is 1 yard and the width is 2 feet. What is the ratio of
length to width of this rectangle?
Solution The ratio must be in terms of the same measure.
1 yd
2 ft
ft
3 13 yd
5 32
Answer The ratio of length to width is 3 : 2.
EXAMPLE 2
The ratio of the length of one of the congruent sides of an isosceles triangle to
the length of the base is 5 : 2. If the perimeter of the triangle is 42.0 centimeters,
what is the length of each side?
Solution Let AB and BC be the lengths of the congruent sides of isosceles ABC and
AC be the length of the base.
B
AB : AC 5 5 : 2 5 5x : 2x
Therefore, AB 5 5x,
BC 5 5x,
A
AB 1 BC 1 AC 5 Perimeter
5x 1 5x 1 2x 5 42
5x
5x
and AC 5 2x.
2x
C
AB 5 BC 5 5(3.5)
5 17.5 cm
12x 5 42
x 5 3.5 cm
Check AB 1 BC 1 AC 5 17.5 1 17.5 1 7.0 5 42.0 cm ✔
Answer The sides measure 17.5, 17.5, and 7.0 centimeters.
AC 5 2(3.5)
5 7.0 cm
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59
Proportion
An equation that states that two ratios are equal is called a proportion. For
example, 3 : 12 5 1 : 4 is a proportion. This proportion can also be written as
3
1
12 5 4.
In general, if b 0 and d 0, then a : b 5 c : d or ba 5 dc are proportions. The
first and last terms, a and d, are called the extremes of the proportion and the
second and third terms, b and c, are the means of the proportion.
If we multiply both sides of the proportion by the product of the second and
last terms, we can prove a useful relationship among the terms of a proportion.
a
b
5 dc
bd A ba B 5 bd A dc B
1
1
bd ¢ ba ≤ 5 bd ¢ dc ≤
1
1
da 5 bc
In any proportion, the product of the means is equal to the product of the
extremes.
EXAMPLE 3
In the junior class, there are 24 more girls than boys. The ratio of girls to boys is
5 : 4. How many girls and how many boys are there in the junior class?
Solution
How to Proceed
(1) Use the fact that the number of girls is
24 more than the number of boys to
represent the number of girls and of
boys in terms of x:
(2) Write a proportion. Set the ratio of the
number of boys to the number of girls, in
terms of x, equal to the given ratio:
(3) Use the fact that the product of the
means is equal to the product of the
extremes. Solve the equation:
Let x 5 the number of boys,
x 1 24 5 the number of girls.
x 1 24
x
5 54
5x 5 4(x 1 24)
5x 5 4x 1 96
x 5 96
x 1 24 5 96 1 24
5 120
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Alternative (1) Use the given ratio to represent the
number of boys and the number of girls
Solution
in terms of x:
Let 5x 5 the number of girls
(2) Use the fact that the number of girls
is 24 more than the number of boys to
write an equation. Solve the equation
for x:
(3) Use the value of x to find the number
of girls and the number of boys:
4x 5 the number of boys
5x 5 24 1 4x
x 5 24
5x 5 5(24) 5 120 girls
4x 5 4(24) 5 96 boys
Answer There are 120 girls and 96 boys in the junior class.
Exercises
Writing About Mathematics
1. If ba 5 dc , then is ac 5 db ? Justify your answer.
d
b
2. If ba 5 dc , then is a 1
5c1
b
d ? Justify your answer.
Developing Skills
In 3–10, write each ratio in simplest form.
3. 12 : 8
7. 6a : 9a, a 0
4. 21 : 14
8.
18
27
5. 3 : 18
9.
24
72
6. 15 : 75
10x
10. 35x
,x0
In 11–19, solve each proportion for the variable.
6
11. x8 5 24
2 1
12. x 15
5 25
2 2
5
13. a 2a
5 14
6
14. y 1 8 5 15
1 3
1 1
15. 3x 16
5 2x 10
x 1 2
2
16. x 2
2
1 5
8
17. 4x 32 8 5 x 2
3
3
2
18. x 2
5x1
2
x
4
19. x5 5 xx 11 13
y 1 3
Applying Skills
20. The ratio of the length to the width of a rectangle is 5 : 4. The perimeter of the rectangle is
72 inches. What are the dimensions of the rectangle?
21. The ratio of the length to the width of a rectangle is 7 : 3. The area of the rectangle is 336
square centimeters. What are the dimensions of the rectangle?
22. The basketball team has played 21 games. The ratio of wins to losses is 5 : 2. How many
games has the team won?
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61
23. In the chess club, the ratio of boys to girls is 6 : 5. There are 3 more boys than girls in the
club. How many members are in the club?
24. Every year, Javier makes a total contribution of $125 to two local charities. The two donations are in the ratio of 3 : 2. What contribution does Javier make to each charity?
25. A cookie recipe uses flour and sugar in the ratio of 9 : 4. If Nicholas uses 1 cup of sugar,
how much flour should he use?
26. The directions on a bottle of cleaning solution suggest that the solution be diluted with
water. The ratio of solution to water is 1 : 7. How many cups of solution and how many cups
of water should Christopher use to make 2 gallons (32 cups) of the mixture?
2-6 COMPLEX RATIONAL EXPRESSIONS
A complex fraction is a fraction whose numerator, denominator, or both contain
fractions. Some examples of complex fractions are:
1
3
2
5
2
7
234
1
8
A complex fraction can be simplified by multiplying by a fraction equal to
1; that is, by a fraction whose non-zero numerator and denominator are equal.
The numerator and denominator of this fraction should be a common multiple
of the denominators of the fractional terms. For example:
1
1
3
3
5
3 33 5 16
2
2
5
5 25 3 77 5 35
2
2
7
7
2 1 34
234
5
3 88 5 16 11 6 5 22
1
1
8
8
A complex fraction can also be simplified by dividing the numerator by the
denominator.
1
3
5 13 4 21 5 13 3 12 5 16
2
5
5 51 4 27 5 5 3 72 5 35
2
2
7
813
11
234
4
4 5 4 5 11 4 1 5 11 3 8 5 88 5 22
5
4
8
4
4
1
1
1
1
8
8
8
A complex rational expression has a rational expression in the numerator,
the denominator, or both. For example, the following are complex rational
expressions.
1
a
a
x13
1
x
1
2
11b
2 b2
1
b
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Like complex fractions, complex rational expressions can be simplified by
multiplying numerator and denominator by a common multiple of the denominators in the fractional terms.
1
1
a
a a
a 5 a ?a
5 a12
(a 2 0)
x13
3 x
5 x1
?x
1
1
x
x
2
5 x 11 3x
12 2
12 2
11b
11b
b2
b2 b2
5
? b2
1
1
12b
12b
2
b 2 2
5 b b12 2
b
5 x2 1 3x
(x 2 0)
5
(b 1 2)(b 2 1)
b(b 2 1)
5
(b 1 2)(b 2 1)
b(b 2 1)
1
1
2
5b1
(b 2 0, 1)
b
Alternatively, a complex rational expression can also be simplified by dividing the numerator by the denominator. Choose the method that is easier for
each given expression.
Procedure
To simplify a complex fraction:
METHOD 1
Multiply by m
m, where m is the least common multiple of the denominators of the
fractional terms.
METHOD 2
Multiply the numerator by the reciprocal of the denominator of the fraction.
EXAMPLE 1
2
Express a3 in simplest form.
4a2
Solution METHOD 1
Multiply numerator and denominator of the fraction by the least common
multiple of the denominators of the fractional terms. The least common multiple of a and 4a2 is 4a2.
2
2
2
a
8a
5 a3 ? 4a
4a2 5 3
3
4a2
4a2
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Complex Rational Expressions
METHOD 2
Write the fraction as the numerator divided by the denominator. Change the
division to multiplication by the reciprocal of the divisor.
2
a
a 5 2 4 3 5 2 ? 4a2 5 2 ? 4a2 5 8a
4a2
3
a
a 3
a 3
3
1
4a2
Answer
8a
3
(a 0)
EXAMPLE 2
b
1
12
Simplify: 10
b2 2 1
25
Solution
METHOD 1
The least common multiple of 10, 2,
and 25 is 50.
b
b
1
1
10 1 2
10 1 2 50
5
? 50
b2
b2
25 2 1
25 2 1
5 5b2 1 25
2b 2 50
5(b 1 5)
5 2(b 1 5)(b 2 5)
5 2(b 52 5)
Answer
5
2(b 2 5)
METHOD 2
b
5
b
1
10 1 2
10 1 10
5
b2
b2
25
25 2 1
25 2 25
b15
10
5 b2 2
25
25
2 2 25
1 5
5 b 10
4 b 25
1
5
1 5
5 b 10
? (b 1 5)25(b 2 5)
2
5
1
5
2(b 2 5)
(b 25, 5)
Exercises
Writing About Mathematics
1
12a
1. For what values of a is A 1 2 a1 B 4 A 1 2 a12 B 5
undefined? Explain your answer.
1 2 a12
d
3
1
2. Bebe said that since each of the denominators in the complex fraction 4 d52 is a non-zero
22 2
constant, the fraction is defined for all values of d. Do you agree with Bebe? Explain why or
why not.
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Developing Skills
In 3–20, simplify each complex rational expression. In each case, list any values of the variables for
which the fractions are not defined.
2
3. 33
7
4. 54
4
5. 83
14
1
2x
1
7. 61 41
312
111
8. a 1 a1
222
9. 1 d
d21
b21
10. 1 b
b21
323
11. b 2 b1
1
2
6. x1
y11
12. 2y 121
15.
18.
13.
a 2 49
a
16.
a 2 9 1 14
a
1 1 y1 2 y62
24
1 1 11
y 1 y2
1
4y 2 y
y 2 12
9
32x
x 2 8 1 15
x
1
1 1 1
14. 2x 1 3x
x2
17.
31 2
11b
b2
1
12 2
b
5 2 452
20. 3 a
a21
21
19. a 1
12a
In 21–24, simplify each expression. In each case, list any values of the variables for which the fractions are not defined.
a
1
7a
11x
3 2
21. 2x
x11
1 8
3
22. 6a42 3a
2 1 a
2
10
5
3
23. A a3 1 a52 B 4 A 10
a 1 6B 1 4
b
12
24. A 6 1 12
b B 4 A 3b 2 b B 1 2 2 b
2-7 SOLVING RATIONAL EQUATIONS
An equation in which there is a fraction in one or more terms can be solved in
different ways. However, in each case, the end result must be an equivalent
equation in which the variable is equal to a constant.
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Solving Rational Equations
65
For example, solve: x4 2 2 5 x2 1 12. This is an equation with rational coefficients and could be written as 14x 2 2 5 12x 1 12. There are three possible methods of solving this equation.
METHOD 1
Work with the fractions as given.
Combine terms containing the variable
on one side of the equation and
constants on the other. Find common
denominators to add or subtract
fractions.
METHOD 2
Rewrite the equation without fractions.
Multiply both sides of the equation by
the least common denominator. In this
case, the LCD is 4.
x
4 2 2
x
x
4 2 2
x
2x
4 2 4
2x4
5 x2 1 12
5 12 1 2
5 12 1 42
5 52
x 5 52 (24) 5 220
2 5 210
x
4
2 2 5 x2 1 12
4 A x4 2 2 B 5 4 A x2 1 12 B
x 2 8 5 2x 1 2
2x 5 10
x 5 210
METHOD 3
Rewrite each side of the equation as a
ratio. Use the fact that the product of
the means is equal to the product of the
extremes.
x
4 2 2
x
8
4 2 4
x 2 8
4
5 x2 1 12
5 x2 1 12
1
5x1
2
4(x 1 1) 5 2(x 2 8)
4x 1 4 5 2x 2 16
2x 5 220
x 5 210
These same procedures can also be used for a rational equation, an equation in which the variable appears in one or more denominators.
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EXAMPLE 1
3
Solve the equation x1 1 13 5 2x
2 1.
Solution Use Method 2.
How to Proceed
(1) Write the equation:
Check
3
1 13 5 2x
21
(2) Multiply both sides of the equation by
the least common denominator, 6x:
1
x
6x A x1
1
(3) Simplify:
1
3B
5
3
6x A 2x
2 1B
3
1
1
x 1 3 5 2x 2 1
?
1
1 13 5 3 2 1
3
2Q 38 R
8
A 1 3 83 B
8
3
?
1 13 5 A 3 3 43 B 2 1
?
1 13 5 4 2 1
9 ?
35
6x
x
18x
1 6x
3 5 2x 2 6x
6 1 2x 5 9 2 6x
3
353✔
(4) Solve for x:
8x 5 3
x 5 38
Answer x 5 38
When solving a rational equation, it is important to check that the fractional
terms are defined for each root, or solution, of the equation. Any solution for
which the equation is undefined is called an extraneous root.
EXAMPLE 2
2
1 2
5 a 2a
Solve for a: a 2
a
Solution Since this equation is a proportion, we can use that the product of the means
is equal to the product of the extremes. Check the roots.
a 2 2
a
1 2
5 a 2a
2a(a 2 2) 5 a(a 1 2)
2a2 2 4a 5 a2 1 2a
a2 2 6a 5 0
a(a 2 6) 5 0
a50
a2650
a56
Answer a 5 6
Check: a 5 0
a 2 2
1 2
5 a 2a
a
0 2 2 ? 0 1 2
0 5 2(0)
Each side of the
equation is undefined
for a 5 0. Therefore,
a 5 0 is not a root.
Check: a 5 6
1 2
a 2 2
5 a 2a
a
6 2 2 ? 6 1 2
6 5 2(6)
4 ? 8
6 5 12
2
2
3 5 3✔
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67
EXAMPLE 3
x
3
4
Solve: x 1
2 5 x 1 x(x 1 2)
Solution Use Method 2 to rewrite the equation without fractions. The least common
denominator is x(x 1 2).
x(x 1
x
x 1 2
x
2) A x 1
2B
5 x3 1 x(x 41 2)
5 x(x 1 2) A x3 B 1 x(x 1 2) A x(x 41 2) B
1
1
1
1
x
3
4
x(x 1 2) a x 1
2 b 5 x (x 1 2) a x b 1 x (x 1 2) a x (x 1 2) b
1
1
1
1
2
x 5 3x 1 6 1 4
2
x 2 3x 2 10 5 0
(x 2 5)(x 1 2) 5 0
x2550
x1250
x55
x 5 22
Check the roots:
Check: x 5 5
x
x 1 2
Check: x 5 2 2
5 x3 1 x(x 41 2)
? 3
5
4
5 1 2 5 5 1 5(5 1 2)
5
5 ? 3
7
4
7 3 5 5 5 3 7 1 35
25 ? 21
4
35 5 35 1 35
25
25
35 5 35 ✔
x
x 1 2
5 x3 1 x(x 41 2)
? 3
22
22 1 2 5 22
22 ?
0 5
1 22(224 1 2)
4
232 1 22(0)
Since x 5 22 leads to a statement
that is undefined, 22 is not a root
of the equation.
Since x 5 5 leads to a true statement,
5 is a root of the equation.
Answer 5 is the only root of x
x
1 2
5 x3 1 x(x 41 2) .
It is important to check each root obtained by multiplying both members of
the given equation by an expression that contains the variable. The derived
equation may not be equivalent to the given equation.
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EXAMPLE 4
Solve and check: 0.2y 1 4 5 5 2 0.05y
Solution The coefficients 0.2 and 0.05 are decimal fractions (“2 tenths” and “5 hundredths”). The denominator of 0.2 is 10 and the denominator of 0.05 is 100.
Therefore, the least common denominator is 100.
0.2y 1 4 5 5 2 0.05y
Check
0.2y 1 4 5 5 2 0.05y
100(0.2y 1 4) 5 100(5 2 0.05y)
?
20y 1 400 5 500 2 5y
0.2(4) 1 4 5 5 2 0.05(4)
25y 5 100
0.8 1 4 5 5 2 0.2
?
y54
4.8 5 4.8 ✔
Answer y 5 4
EXAMPLE 5
On his way home from college, Daniel traveled 15 miles on local roads and 90 miles on
the highway. On the highway, he traveled 30
miles per hour faster than on local roads. If
the trip took 2 hours, what was Daniel’s rate
of speed on each part of the trip?
Solution Use distance
rate 5 time to represent the time for each part of the trip.
Let x 5 rate on local roads. Then 15
x 5 time on local roads.
90
Let x 1 30 5 rate on the highway. Then x 1
30 5 time on the highway.
The total time is 2 hours.
90
1x1
30 5 2
90
15
x(x 1 30) A x B 1 x(x 1 30) A x 1 30 B 5 2x(x 1 30)
15
x
15(x 1 30) 1 90x 5 2x2 1 60x
15x 1 450 1 90x 5 2x2 1 60x
0 5 2x2 2 45x 2 450
Recall that a trinomial can be factored by writing the middle term as the sum of
two terms whose product is the product of the first and last term.
2x2(2450) 5 2900x2
Find the factors of this product whose sum is the middle term 245x.
15x 1 (260x) 5 245x
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69
Write the trinomial with four terms, using this pair of terms in place of 245x,
and factor the trinomial.
0 5 2x2 2 45x 2 450
0 5 2x2 1 15x – 60x 2 450
0 5 x(2x 1 15) 2 30(2x 1 15)
0 5 (2x 1 15)(x 2 30)
2x 1 15 5 0
2x 5 215
x5
x 2 30 5 0
x 5 30
215
2
Reject the negative root since a rate cannot be a negative number. Use x 5 30.
1
On local roads, Daniel’s rate is 30 miles per hour and his time is 15
30 5 2 hour.
On the highway, Daniel’s rate is 30 1 30 or 60 miles per hour. His time is
90
3
3
1
4
60 5 2 hours. His total time is 2 1 2 5 2 5 2 hours.
Answer Daniel drove 30 mph on local roads and 60 mph on the highway.
Exercises
Writing About Mathematics
2
1 2
1. Samantha said that the equation a 2
in Example 2 could be solved by multiplying
5 a 2a
a
both sides of the equation by 2a. Would Samantha’s solution be the same as the solution
obtained in Example 2? Explain why or why not.
3
5
x 2 2
2
5x1
2. Brianna said that x 2
2 5 x 1 2 is a rational equation but 3
5 is not. Do you
agree with Brianna? Explain why or why not.
Developing Skills
In 3–20, solve each equation and check.
3. 14a 1 8 5 12a
x
6. x5 2 10
57
4. 34x 5 14 2 x
2
2
5x2
5. x 1
3
5
8. x 2 0.05x 5 19
3x
7. 2x
3 1 1 5 4
10. 0.2a 5 0.05a 1 3
11. 1.2b 2 3 5 7 2 0.05b
x
2
12. x 1
5 5 3
13. 2x 72 3 5 x4
11
14. a3 1 12 5 5a
15. 18 2 b4 5 10
16. x2 5 2x 31 1
5
5
17. 1 5 x 1
3 1 (x 1 2)(x 1 3)
8
1
5a1
18. a 2
4
3
8
4
19. y 1
2 5 1 2 y(y 1 2)
20. 3b 42 2 2 3b 71 2 5 2 1
9b 2 4
9. 0.4x 1 8 5 0.5x
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Applying Skills
21. Last week, the ratio of the number of hours that Joseph worked to the number of hours that
Nicole worked was 2 : 3. This week Joseph worked 4 hours more than last week and Nicole
worked twice as many hours as last week. This week the ratio of the hours Joseph worked to
the number of hours Nicole worked is 1 : 2. How many hours did each person work each
week?
22. Anthony rode his bicycle to his friend’s house, a distance of 1 mile. Then his friend’s mother
drove them to school, a distance of 12 miles. His friend’s mother drove at a rate that is 25
miles per hour faster than Anthony rides his bike. If it took Anthony 35 of an hour to get to
school, at what average rate does he ride his bicycle? (Use distance
rate 5 time for each part of
the trip to school.)
23. Amanda drove 40 miles. Then she increased her rate of speed by 10 miles per hour and
drove another 40 miles to reach her destination. If the trip took 145 hours, at what rate did
Amanda drive?
24. Last week, Emily paid $8.25 for x pounds of apples. This week she paid $9.50 for (x 1 1)
pounds of apples. The price per pound was the same each week. How many pounds of
apples did Emily buy each week and what was the price per pound? (Use
total cost
number of pounds 5 cost per pound for each week.)
2-8 SOLVING RATIONAL INEQUALITIES
Inequalities are usually solved with the same procedures that are used to solve
equations. For example, we can solve this equation and this inequality by using
the same steps.
1 18 5 13x 1 38
24 A 14x B 1 24 A 18 B 5 24 A 13x B 1 24 A 38 B
6x 1 3 5 8x 1 9
1
4x
22x 5 6
x 5 23
1 18 , 13x 1 38
24 A 14x B 1 24 A 18 B , 24 A 13x B 1 24 A 38 B
6x 1 3 , 8x 1 9
1
4x
22x , 6
x . 23
All steps leading to the solution of this equation and this inequality are the
same, but special care must be used when multiplying or dividing an inequality.
Note that when we multiplied the inequality by 24, a positive number, the order
of the inequality remained unchanged. In the last step, when we divided the
inequality by 22, a negative number, the order of the inequality was reversed.
When we solve an inequality that has a variable expression in the denominator by multiplying both sides of the inequality by the variable expression, we
must consider two possibilities: the variable represents a positive number or the
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Solving Rational Inequalities
71
variable represents a negative number. (The expression cannot equal zero as
that would make the fraction undefined.)
9
x
For example, to solve 2 2 x 1
1 . 3 2 x 1 1, we multiply both sides of the
inequality by x 1 1. When x 1 1 is positive, the order of the inequality remains
unchanged. When x 1 1 is negative, the order of the inequality is reversed.
If x 1 1 0, then x –1.
22
9
x 1 1
2
x
x 1 1
x
x 1 1
.32
If x 1 1 0, then x –1.
9
x 1 1
22x
9
x 1 1 2 x
.322
1
x
1 1
x
1 1
9
.32x1
1
.322
1
x
(x 1 1) 9x 2
1 1 . (x 1 1)(1)
x
(x 1 1) 9x 2
1 1 , (x 1 1)(1)
1
1
92x.x11
22x
22
92x,x11
, 28
22
22x
22
x,4
. 28
22
x.4
Therefore, x 21 and x 4 or
21 x 4.
There are no values of x such
that x 21 and x 4.
9
x
The solution set of the equation 2 2 x 1
1 . 3 2 x 1 1 is {x : 1 x 4}.
An alternative method of solving this inequality is to use the corresponding
equation.
STEP 1. Solve the corresponding equation.
9
x
22x1
1 5 3 2 x 1 1
1
1
9
x
(x 1 1)(2) 2 (x 1 1) x 1
1 5 (x 1 1)(3) 2 (x 1 1) x 1 1
1
1
2x 1 2 2 x 5 3x 1 3 2 9
22x 5 28
x54
STEP 2. Find any values of x for which the equation is undefined.
9
x
Terms x 1
1 and x 1 1 are undefined when x 5 21.
STEP 3. To find the solutions of the corresponding inequality, divide the num-
ber line into three intervals using the solution to the equality, 4, and the
value of x for which the equation is undefined, 21.
3 2 1 0
1
2
3
4
5
6
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The Rational Numbers
Choose a value from each section and substitute that value into the inequality.
3 2 1 0
1
2
3
4
5
6
Let x 5 22:
Let x 5 0:
Let x 5 5:
9
x
22x1
1 . 3 2 x 1 1
9
x
22x1
1 . 3 2 x 1 1
9
x
22x1
1 . 3 2 x 1 1
?
?
9
0
2201
1 . 3 2 0 1 1
?
9
5
2251
1 . 3 2 5 1 1
?
?
2 2 56 . 3 2 96
9
7
666✘
9
2 2 2222
1 1 . 3 2 22 1 1
?
2 2 (2) . 3 2 (29)
2 2 (0) . 3 2 (9)
0 6 12 ✘
2 . 26 ✔
The inequality is true for values in the interval 21 x 4 and false for all
other values.
EXAMPLE 1
Solve for a: 2 2 a3 . a5
Solution METHOD 1
Multiply both sides of the equation by a. The sense of the inequality will
remain unchanged when a 0 and will be reversed when a 0.
Let a 0:
aA2 2
3
aB
.
Let a 0:
a A a5 B
a A 2 2 a3 B , a A a5 B
2a 2 3 , 5
2a 2 3 . 5
2a . 8
2a , 8
a.4
a,4
a 0 and a 4 → a 4
a 0 and a 4 → a 0
The solution set is {a : a 0 or a 4}.
METHOD 2
Solve the corresponding equation for a.
2 2 a3 5 a5
a A 2 2 a3 B 5 a A a5 B
2a 2 3 5 5
2a 5 8
a54
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Solving Rational Inequalities
73
Partition the number line using the solution to the equation, a 5 4, and the
value of a for which the equation is undefined, a 5 0. Check in the inequality
a representative value of a from each interval of the graph.
3 2 1 0
1
2
3
4
5
6
Let a 5 21:
Let a 5 1:
Let a 5 5:
3
a
3
a
5
a
2 2 a3 . a5
2 2 31 . 51
2 2 35 . 55
22
.
5
a
22
?
.
?
3
5
2 2 21
. 21
?
?
2 1 3 . 25
223.5
5 . 25 ✔
21 6 5 ✘
?
10
5
?
2 35 . 1
7
5
.1✔
Answer {a : a 0 or a 4}
Exercises
Writing About Mathematics
5
1. When the equation 2 2 b3 5 b 1
2 is solved for b, the solutions are 21 and 3. Explain why
the number line must be separated into five segments by the numbers 22, 21, 0, and 3 in
5
order to check the solution set of the inequality 2 2 b3 . b 1
2.
x
2. What is the solution set of x , 0? Justify your answer.
Developing Skills
In 3–14, solve and check each inequality.
y 2 3
5
y 1 2
10
3. a4 . a2 1 6
4.
2
d
.d2
6. 2 2
7
5
1
2 2 . 11 2 a6
7. a 1
4
9. 5 2 y7 , 2 1 y5
12. x2 1 x3 , 10
,
5. 3b 82 4 , 4b 42 3
7
2 x2 . 32
8. 2x
2
10. 3 2 a 1
1 , 5
2
11. x 4x
2 4 1 2 , x 2 4
x
1
13. x 1
5 2 x 1 5 . 4
9
1
14. 3 2 a 1
1 . 5 2 a 1 1
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The Rational Numbers
CHAPTER SUMMARY
A rational number is an element of the set of numbers of the form ba when
a and b are integers and b 0. For every rational number ba that is not equal to
zero there is a multiplicative inverse or reciprocal ba such that ba ? ba 5 1.
A number is a rational number if and only if it can be written as an infinitely
repeating decimal.
A rational expression is the quotient of two polynomials. A rational expression has no meaning when the denominator is zero.
A rational expression is in simplest form or reduced to lowest terms when
there is no factor of the numerator that is also a factor of the denominator
except 1 and 21.
If ba and dc are two rational numbers with b 0 and d 0:
a
b
ac
? dc 5 bd
a
b
4 dc 5 ba ? dc 5 ad
bc (c 0)
a
b
ad
cb
1 cb
1 dc 5 ba ? dd 1 dc ? bb 5 bd
1 bd
5 ad bd
A ratio is the comparison of two numbers by division. The ratio of a to b can
be written as ba or as a : b when b 0.
An equation that states that two ratios are equal is called a proportion. In
the proportion a : b 5 c : d or ba 5 dc , the first and last terms, a and d, are called
the extremes, and the second and third terms, b and c, are the means.
In any proportion, the product of the means is equal to the product of the
extremes.
A complex rational expression has a rational expression in the numerator,
the denominator or both. Complex rational expressions can be simplified by
multiplying numerator and denominator by a common multiple of the denominators in the numerator and denominator.
A rational equation, an equation in which the variable appears in one or
more denominators, can be simplified by multiplying both members by a common multiple of the denominators.
When a rational inequality is multiplied by the least common multiple of
the denominators, two cases must be considered: when the least common multiple is positive and when the least common multiple is negative.
VOCABULARY
2-1 Rational number • Multiplicative inverse • Reciprocal
2-2 Rational expression • Simplest form • Lowest terms
2-4 Least common denominator (LCD) • Least common multiple (LCM)
2-5 Ratio • Ratio in simplest form • Proportion • Extremes • Means
2-6 Complex fraction • Complex rational expression
2-7 Rational equation • Extraneous root
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Review Exercises
75
REVIEW EXERCISES
1. What is the multiplicative inverse of 72?
2(b 1 1)
2. For what values of b is the rational expression b(b2 2 1) undefined?
5
3. Write 12
as an infinitely repeating decimal.
In 4–15, perform the indicated operations, express the answer in simplest form,
and list the values of the variables for which the fractions are undefined.
3
10a2b
4. 5ab
2 ?
9
1
2
1 4a
5. 5a
1 12
? x2 22 16
6. 3x 4x
2
2
2a
1 10
4 a 1 7a
7. a 1
5a
a2
3
2
8. a 1
1 1 a2 2 1
9. y 2 3 2 y2 18
2 9
y
2
2 18
12
10. d 1 3d
4 2d 1
8
4d
3
10
11. 5b
? 5b 1
2 b1
6
3
16 2 a2
1
13. A a 2
4 1 4 2 aB ?
2
a
a
1
12. a 1
2 1 a2 1 a 4 a 1 1
3
1
14. A 1 1 x 1
1 B ? A 1 1 x2 2 4 B
15. A x 2 x1 B 4 (1 2 x)
In 16–19, simplify each complex rational expression and list the values of the
variables (if any) for which the fractions are undefined.
16.
1 1 21
2 1 14
17.
a1b
b
11b
a
x
1
22
18. 12
2
x
19.
12 2 3
a 2 1 2 20
a
1 2 16
2
a
In 20–27, solve and check each equation or inequality.
3
5 5a
20. a7 1 14
2
21. x5 1 9 5 12 2 x4
22. y3 1 12 5 y6
3x
23. x 2x
2 2 5 x 1 1
x
6
5
24. x 2
3 1 x(x 2 3) 5 x 2 3
25. 2d 31 1 5 2d d1 2 2 2
26. x1 1 3 . x7
27. 2x 11 1 2 2 # 8
28. The ratio of boys to girls in the school chorus was 4 : 5. After three more
boys joined the chorus, the ratio of boys to girls is 9 : 10. How many boys
and how many girls are there now in the chorus?
29. Last week Stephanie spent $10.50 for cans of soda. This week, at the same
cost per can, she bought three fewer cans and spent $8.40. How many cans
of soda did she buy each week and what was the cost per can?
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30. On a recent trip, Sarah traveled 80 miles at a constant rate of speed. Then
she encountered road work and had to reduce her speed by 15 miles per
hour for the last 30 miles of the trip. The trip took 2 hours. What was
Sarah’s rate of speed for each part of the trip?
31. The areas of two rectangles are 208 square feet and 182 square feet.
The length of the larger is 2 feet more than the length of the smaller. If
the rectangles have equal widths, find the dimensions of each rectangle.
area
5 width. B
A Use length
Exploration
The early Egyptians wrote fractions as the sum of unit fractions (the reciprocals
of the counting numbers) with no reciprocal repeated. For example:
3
4
2
1
1
1
1
1
5 24 1 14 5 12 1 14
and
3 5 3 1 3 5 3 1 4 1 12
Of course the Egyptians used hieroglyphs instead of the numerals familiar
to us today. Whole numbers were represented as follows.
1
2
3
4
5
6
7
8
9
or
10
100
1,000
100,000
10,000
To write unit fractions, the symbol
hieroglyph. For example:
5
1
3
1,000,000
10,000,000
was drawn over the whole number
1
5 20
and
1
1
1. Show that for n 0, n1 5 n 1
1 1 n(n 1 1) .
2. Write 34 and 23 using Egyptian fractions.
3. Write each of the fractions as the sum of unit fractions. (Hint: Use the
results of Exercise 1.)
a. 52
7
b. 10
7
c. 12
23
d. 24
e. 11
18
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Cumulative Review
CUMULATIVE REVIEW
77
CHAPTERS 1–2
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. Which of the following is not true in the set of integers?
(1) Addition is commutative.
(2) Every integer has an additive inverse.
(3) Multiplication is distributive over addition.
(4) Every non-zero integer has a multiplicative inverse.
2. 2 2 3 2 5 is equal to
(1) 0
(2) 2
(3) 6
(4) 4
3. The sum of 3a2 2 5a and a2 1 7a is
(1) 3 1 2a
(2) 3a2 1 2a
(3) 4a2 1 2a
(4) 6a2
4. In simplest form, 2x(x 1 5) 2 (7x 1 1) is equal to
(1) 2x2 1 3x 1 1
(3) 2x2 1 12x 1 1
(2) 2x2 1 3x 2 1
(4) 2x2 2 7x 1 4
5. The factors of 2x2 2 x 2 6 are
(1) (2x 2 3)(x 1 2)
(2) (2x 1 2)(x 2 3)
(3) (2x 1 3)(x 2 2)
(4) (2x 2 3)(x 2 2)
6. The roots of the equation x2 2 7x 1 10 5 0 are
(1) 2 and 5
(3) 27 and 10
(2) 22 and 25
(4) 25 and 2
2
21
7. For a 0, 2, 22, the fraction a 4 is equal to
a2a
1
(1) a 1
2
1
(2) 2a 1
2
1
(3) a 2
2
1
(4) 2a 2
2
8. In simplest form, the quotient 2b b2 1 4 8b 3b
2 4 equals
(1) 34
(2) 34
2
(3) 4(2b3b2 1) 2
(4) 4(2b7a2 1)
9. For what values of a is the fraction a2 22a2a1 24 35 undefined?
(1) 22, 25, 7
(2) 2, 5, 27
(3) 25, 7
(4) 5, 27
10. The solution set of the equation 2x 2 1 5 7 is
(1) {23, 4}
(2) (3) {23}
(4) {4}
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Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Solve and check: 7 2 2x 3
12. Perform the indicated operations and write the answer in simplest form:
3
a 1 5
2 2 9
3
2a2
4 a 15
5
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Write the following product in simplest form and list the values of x for
1 5 x2 2 x
which it is undefined: 5x
15
x2 2 1 ?
14. The length of a rectangle is 12 meters longer than half the width. The area
of the rectangle is 90 square meters. Find the dimensions of the rectangle.
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. Find the solution set and check: 2x2 2 5x 7
16. Diego had traveled 30 miles at a uniform rate of speed when he encountered construction and had to reduce his speed to one-third of his original
rate. He continued at this slower rate for 10 miles. If the total time for
these two parts of the trip was one hour, how fast did he travel at each
rate?
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CHAPTER
REAL NUMBERS
AND RADICALS
Traditionally, we use units such as inches or meters
to measure length, but we could technically measure
with any sized unit. Two line segments are called commeasurable if there exists some unit with which both
segments have integral measures. Therefore, the ratio
of the lengths of the line segments is the ratio of integers, a rational number. Two line segments are incommensurable if there is no unit, no matter how small,
with which both segments can have integral measures.
Therefore, the ratio of the lengths is not a rational
number, that is, the ratio is an irrational number.
The oldest known proof of the existence of incommensurable line segments is found in the tenth book of
Euclid’s Elements although the proof was known long
before Euclid’s time. Euclid established that the diagonal of a square and a side of a square are incommensurable. The ratio of their lengths is an irrational
number. In arithmetic terms, what Euclid proved was
that !2 was irrational.The concept of a real number,
then, had its beginnings as the ratio of the lengths of
line segment.
3
CHAPTER
TABLE OF CONTENTS
3-1 The Real Numbers and
Absolute Value
3-2 Roots and Radicals
3-3 Simplifying Radicals
3-4 Adding and Subtracting
Radicals
3-5 Multiplying Radicals
3-6 Dividing Radicals
3-7 Rationalizing a Denominator
3-8 Solving Radical Equations
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
79
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3-1 THE REAL NUMBERS AND ABSOLUTE VALUE
In Chapter 2 we learned that a number is rational if and only if it can be written as an infinitely repeating decimal. However, we know that there are infinite
decimal numbers that do not repeat. For example, the following numbers have
patterns that continue infinitely but do not repeat.
0.20200200020000200000 . . .
0.123456789101112131415 . . .
These numbers are called irrational numbers.
DEFINITION
An irrational number is an infinite decimal number that does not repeat.
Other examples of irrational numbers are numbers that are the square root
of an integer that is not a perfect square. For example, !2, !3, !5, and !7 are
all irrational numbers. The ratio of the circumference of a circle to its diameter,
p, is also an irrational number.
DEFINITION
The union of the set of rational numbers and the set of irrational numbers is
the set of real numbers.
Every point on the number line is associated with one and only one real
number and every real number is associated with one and only one point on
the number line. We say that there is a one-to-one correspondence between the
real numbers and the points on the number line.
EXAMPLE 1
Which of the following numbers is irrational?
27, 54, 2.154, 0.12131415 . . .
5
Solution 27 5 27
1 and 4 are each the ratio of integers and are therefore rational.
2.154 5 2.1545454 . . . is an infinite repeating decimal and is therefore rational.
0.12131415 . . . is an infinite decimal that does not repeat and is therefore irrational.
Answer 0.12131415 . . .
Graphing Inequalities on the Number Line
In the set of integers, the solution set of an inequality can often be listed.
For example, when the domain is
the set of integers, the solution set of
the inequality shown at the right is
{21, 0, 1, 2, 3, 4, 5}.
5,x17
# 12
5 2 7 , x 1 7 2 7 # 12 2 7
22 , x
# 5
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The Real Numbers and Absolute Value
If the domain is the set of real numbers, the elements of the solution set cannot
be listed but can be shown on the number line.
3 2 1 0
1
2
3
4
5
6
7
Note 1: The open circle at 22 indicates that this is a lower boundary of the set
but not an element of the set. The darkened circle at 5 indicates that 5 is the
upper boundary of the set and an element of the set.
Note 2: The set can be symbolized using interval notation as (22, 5]. The curved
parenthesis, (, at the left indicates that the lower boundary is not an element of
the solution set and the bracket, ], at the right indicates that the upper boundary is included in the set. The set includes all real numbers from 22 to 5, including 5 but not including 22.
Absolute Value Inequalities
We know that if x 5 3, x 5 3, or x 5 23. These two points on the real number
line separate the line into three segments.
3
0
3
• The points between 23 and 3 are closer
to 0 than are 23 and 3 and therefore have
absolute values that are less than 3.
• The points to the left of 23 and to the right
of 3 are farther from 0 than are 23 and 3
and therefore have absolute values that are
greater than 3.
3
0
3
x , 3
3
0
3
x . 3
In Chapter 1, we learned that for any positive value of k:
• if x , k, then 2k , x , k.
• if x . k, then x . k or x , 2k.
What about when k is negative? We can use the definition of absolute value
to solve inequalities of the form x , k and x . k.
Solve x , k for negative k.
Since x is always non-negative and a
non-negative number is always greater
than a negative number, there is no
value of x for which x , k is true.
The solution set is the empty set.
Solve x . k for negative k.
Since x is always non-negative and
a non-negative number is always
greater than a negative number,
x . k is always true. The solution
set is the set of real numbers.
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SUMMARY
Inequality
Value of k
Solution Set
|x| , k
k.0
{x : 2k , x , k}
|x| , k
k,0
0
|x| . k
k.0
{x : x , 2k or x . k}
|x| . k
k,0
{All real numbers}
EXAMPLE 2
Solve for x and graph the solution set: 3a – 1 , 5.
Solution For a positive value of k, the solution of x , k is 2k , x , k.
Let x 5 3a 2 1 and k 5 5.
How to Proceed
(1) Write the solution in terms of a:
25 , 3a 2 1
(2) Add 1 to each member of the
inequality:
,5
25 1 1 , 3a 2 1 1 1 , 5 1 1
(3) Divide each member by 3:
(4) Graph the solution set:
24 , 3a
,6
243 , 3a
3
4
23 , a
, 63
43
,2
0
2
Check We can check the solution on the graphing calculator.
Graph Y1 5 3X 2 1 and Y2 5 5. Absolute value is entered in the calculator by using the abs (function ( MATH
1
). By pressing TRACE 24 3 ENTER and TRACE 2 ENTER , we can see that the two functions intersect
at points for which x 5 243 and x 5 2. From the graph, we can verify that A 243, 2 B
is the solution to the inequality since the graph of Y1 lies below the graph of Y2
in this interval.
Y1=abs(3X-1)
Y1=abs(3X-1)
*
*
X=-1.333333
Y=5
X=2
Y=5
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83
Writing About Mathematics
1. Tony said that
3
1 2 51
is irrational because it is not the ratio of integers and is therefore not a
rational number. Do you agree with Tony? Explain why or why not.
2. Maria said that since the solution of the inequality 2x – 5 , 3 can be found by using
23 , 2x 2 5 , 3, then the solution of the inequality 2x – 5 . 3 can be found by using
23 . 2x 2 5 . 3. Do you agree with Maria? Explain why or why not.
Developing Skills
In 3–14, determine whether each of the numbers is rational or irrational.
3. 04
4. 2.17273747 . . .
5. 3p
6. !17
7. 134
8. !2
2
!5
10. !5
11. !16
2
9. !3 1 5
12. 0 1 p
13. p
p
14. 213 1 !3
In 15–26, find and graph the solution set of each inequality.
15. x , 7
16. a 2 5 $ 3
17. 2y 1 5 . 9
18. 2 2 4b # 6
19. –5 2 a . 4
20. 9 2 3x 1 3 . 0
21. P 5x 2 12 P 2 32 . 0
22. 2x 1 2 # 23
23. 2x 1 2 . 23
24. P 52x 1 2 P # 0
25. P x 1 12 P 1 1 . 12
26. 3 2x 2 2 1 2 $ 25
Applying Skills
27. The temperature on Mars roughly satisfies the inequality t 2 75 # 145 where t is the temperature in Fahrenheit. What is the range of temperatures on Mars?
28. Elevation of land in the United States is given by the inequality h 2 10,019 # 10,301 where
h is the height in feet. What is the range of elevations in the United States?
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3-2 ROOTS AND RADICALS
Square Root
We know that 4 5 2 ? 2. Therefore, 4 is the square of 2, and 2 is a square root of
4. We also know that 4 5 22 ? (22). Therefore, 4 is the square of 22, and 22 is
a square root of 4. We write this as 4 5 22 and 4 5 (22)2 . Because there are
two numbers whose square is 4, 4 has two square roots, written !4 5 2 and
2!4 5 22. We can write these last two equalities as 6 !4 5 62.
DEFINITION
A square root of k is one of the two equal factors whose product is k.
Every positive real number has two square roots: a positive real number and
its opposite, a negative real number. For example:
x2 5 9
x 5 6 !9 5 63
x 5 3 or x 5 23
y2 5 5
y 5 6 !5
y 5 !5 or y 5 2!5
The principal square root of a positive real number is its positive square
root. Thus, the principal square root of 9 is 3 and the principal square root of 5
is !5. In general, when referring to the square root of a number, we mean the
principal square root.
When !5 is entered, a calculator will return the approximate decimal
equivalent of !5 to as many decimal places as allowed by the calculator.
ENTER:
DISPLAY:
2nd
¯ 5 ENTER
√(5
2.236067977
If we enter the decimal 2.236067977 into the calculator and square it, the
calculator will return 4.999999998, an approximate value correct to nine decimal
places. If we square 2.236067977 using pencil and paper we obtain a number
with eighteen decimal places with 9 in the last decimal place. The exact decimal
value of !5 is infinite and non-repeating.
Cube Root
DEFINITION
The cube root of k is one of the three equal factors whose product is k.
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Roots and Radicals
85
3
The cube root of k is written as !
k. For example, since 5 ? 5 ? 5 5 125, 5 is
3
one of the three equal factors whose product is 125 and !
125 5 5. There is only
one real number that is the cube root of 125. This real number is called the principal cube root. In a later chapter, we will enlarge the set of real numbers to
form a set called the complex numbers. In the set of complex numbers, every
real number has three cube roots, only one of which is a real number.
3
Note that (25) ? (25) ? (25) 5 2125. Therefore, 25 5 !2125 and 25 is the
principal cube root of 2125.
The nth Root of a Number
DEFINITION
The nth root of k is one of the n equal factors whose product is k.
➛
n
The nth root of k is written as !k. In this notation, k is the radicand, n is the
n
index, and !k is the radical. If k is positive, the principal nth root of k is the positive nth root.
If k is negative and n is odd, the principal nth root of k is negative. If k is
negative and n is even, there is no principal nth root of k in the set of real numbers.
4
For example, in the expression !16:
Index
Radical
4
➛ !16
➛
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Radicand
Since 2 ? 2 ? 2 ? 2 5 16, !16 5 2 and 2 is the principal fourth root of 16. Note
4
that (22) ? (22) ? (22) ? (22) 5 16, so 2!16 5 22. The real number 16 has two
fourth roots that are real numbers: 2 (the principal fourth root) and 22.
If the real number k is not the product of n equal factors that are rational
n
numbers, then the principal nth root of kn, !k, is irrational. For example, since
13 is a prime and has only itself and 1 as factors, the principal nth root of 13,
n
!
13, is an irrational number.
Since the product of an even number of positive factors is positive and the
n
product of an even number of negative factors is positive, !k is a real number
4
4
when n is even if and only if k is positive. For example !81 5 3 but !281 has
no real roots.
4
To evaluate a radical on the calculator, we need to access the MATH menu.
x
3
Entry 4, !
(, is used to evaluate cube root. Entry 5, ! , is used to evaluate a
radical with any index. When using entry 5, the index of the root must be
entered first.
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Evaluate !12
Evaluate !35
3
ENTER:
MATH
DISPLAY:
3
4
12 ENTER
5
ENTER:
DISPLAY:
√(12
5 MATH
5
35 ENTER
5 x√ 3 5
2.289428485
2.036168005
When a variable appears in the radicand, the radical can be simplified if
the exponent of the variable is divisible by the index. For instance, since
3
8x6 5 (2x2)(2x2)( 2x2), "
8x6 5 2x2.
EXAMPLE 1
If a2 5 169, find all values of a.
Solution Since 13 ? 13 5 169 and 213 ? (213) 5 169, a 5 6 !169 5 613. Answer
EXAMPLE 2
Evaluate each of the following in the set of real numbers:
b. 2"121b4
a. !49
3
218
c. #
4
d. !
281
Solution a. 7 ? 7 5 49. Therefore, 7 is the principal square root and !49 5 7.
b. (211b2)(211b2) 5 121b4. Therefore, 11b2 is the principal square root of
"121b4 and 2"121b4 5 211b2.
c. A 212 B A 212 B A 212 B 5 218. Therefore, 212 is the principal cube root of 218 and
#28 5 22.
3
1
1
d. A negative real number does not have four equal factors that are real num4
bers. !
281 does not represent a real number.
Answers a. 7 b. 11b2 c. 212 d. not a real number
EXAMPLE 3
Find the length of the longer leg of a right triangle if the measure of the shorter
leg is 9 centimeters and the measure of the hypotenuse is 41 centimeters.
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Roots and Radicals
87
Solution Use the Pythagorean Theorem, a2 1 b2 5 c2. Let a be the length of the longer
leg, b be the length of the shorter leg, and c be the length of the hypotenuse.
a2 1 b2 5 c2
a2 1 92 5 412
a2 1 81 5 1,681
a2 5 1,600
a 5 640
Reject the negative root.
Answer The length of the longer leg is 40 centimeters.
Exercises
Writing About Mathematics
1. a. Kevin said that if the index of a radical is even and the radicand is positive, then the radical has two real roots. Do you agree with Kevin? Explain why or why not.
b. Kevin said that if the index of a radical is odd, then the radical has one real root and
that the root is positive if the radicand is positive and negative if the radicand is negative. Do you agree with Kevin? Explain why or why not.
2. a. Sarah said that in the set of real numbers, !a is one of the two equal factors whose product is a. Therefore, !a ? !a 5 a for some values of a. Do you agree with Sarah? Explain
why or why not.
b. If you agree with Sarah, for which values of a is the statement true? Explain.
Developing Skills
In 3–10, tell whether each represents a number that is rational, irrational, or neither.
3. !25
7. !0
4. !8
4
8. !
16
5. !28
5
9. !
2243
In 11–38, evaluate each expression in the set of real numbers.
11. !16
15. !169
3
19. !27
3
23. 2!2125
8
27. #27
3
12. 6 !16
16. 2!0.04
4
20. !16
5
24. !21
5
1
28. #232
3
6. !
28
10. !0.25
13. 2!16
14. !625
3
21. !2125
3
22. 2!125
17. 6 !0.64
4
25. #25
3
29. !0.001
18. !1.44
26. 2#49
36
4
30. !0.0256
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Real Numbers and Radicals
31. "x6, x 0
4 b
35. 2#
1,296, b 0
32. "100c4, c 0
8
k
37. !1
, k an integer greater than 2
33. "0.25x2, x 0
3
34. #
a3 , a 0
1,000
5
36. "
20.00001y5, y 0
38. "x2k, k an integer greater than 2
k
In 39–42, find the set of real numbers for which the given radical is a real number.
39. !x 2 2
40. !9 2 3x
41. !4x 1 12
4
42. !
x15
45. b2 5 100
46. y2 2 169 5 0
In 43–46, solve each equation for the variable.
43. x2 5 81
44. a2 5 196
Applying Skills
47. The area of a square is 14 square centimeters. What is the length of a side of the square?
48. Find the length of the hypotenuse of a right triangle if the length of the longer leg is 20 feet
and the length of the shorter leg is 12 feet.
49. Find the length of the shorter leg of a right triangle if the length of the longer leg is 36
inches and the length of the hypotenuse is 39 inches.
50. What is the length of a side of a square if the length of a diagonal is !72 inches?
51. The length l, width w, and height h, of a rectangular carton are l 5 12 feet, w 5 4 feet, and
h 5 3 feet. The length of a diagonal, d, the distance from one corner of the box to the opposite corner, is given by the formula d2 5 l2 1 w2 1 h2. What is the length of the diagonal of
the carton?
3-3 SIMPLIFYING RADICALS
We know that !4 ? 9 5 !36 5 6 and that !4 ? !9 5 2 3 5 6. Therefore,
"4 ? 9 5 "4 ? "9.
We can state this relationship in general terms. For all non-negative real numbers a and b:
!a ? b 5 !a ? !b
This relationship can be used when !a and !b are rational numbers and
when they are irrational numbers. When !a ? b is an irrational number, we can
often write it in simpler form. For instance:
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Simplifying Radicals
!450
!450
5 !9 ? !50
!450
5 !25 ? !18
5 3 !50
89
5 !225 ? !2
5 5 !18
5 15 !2
Each of these irrational numbers is equal to !450. We say that 15 !2 is the
simplest form of !450 because the radicand has no factor that is a perfect
square. (Alternatively, a radical is in simplest form when the radicand is a prime
number or the product of prime numbers with each prime factor occurring
once.) Note that the radicand in each of the other forms of !450 has a factor
that is a perfect square and can be simplified.
!450 5 3 !50
!450 5 5 !18
5 3 !25!2
5 5 !9!2
5 3 ? 5 !2
5 5 ? 3 !2
5 15 !2
5 15 !2
We know that x3 x3 5 x6. Then for x 0, "x6 5 x3. In general, xn xn 5 x2n.
Therefore, for x 0:
"x 2n 5 x n
In particular, when n 5 1, we have that "x2 5 x.
We can also write x5 as x4 x. Therefore,
"x5 5 "x4 ? x
5 "x4 ? !x
5 x2 !x
"12y7 5 "4y6 ? !3y
5 2y3 !3y
(y $ 0)
"50a3b5 5 "25a2b4 ? !2ab
5 5ab2 !2ab
(a $ 0 and b $ 0)
(x $ 0)
Fractional Radicands
When the radicand of an irrational number is a fraction, the radical is in simplest
form when it is written with an integer under the radical sign. For example, #23
does not have a perfect square factor in either the numerator or the denominator of the radicand. The radical is not in simplest form. To simplify this radical,
write 23 as an equivalent fraction with a denominator that is a perfect square.
3
6
6
2
2
1
1
1
#3 5 #3 3 3 5 #9 5 #9 3 1 5 #9 3 !6 5 3 !6 or
!6
3
Note that since ac 5 A 1c B A a1 B 5 A 1c B (a) and the square root of a product is
equal to the product of the square roots of the factors:
a
1
#c 5 #c ?
a
1
1
5 #1c ? #a1 5 !c
!a 5 !a
!c
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Therefore, for any non-negative a and positive c:
a
#c 5
!a
!c
This leads to an alternative method of simplifying a fractional radicand:
5
#7 5
!5
!7
a
#c 5
!a
!c
3 !7
5 !35
5 !35
7
!7
!49
or
3 !c
5 !ac
5 !ac
c
!c2
!c
EXAMPLE 1
Write each expression in simplest form.
a. #89
b. #45
c. #38
Solution a. Here the denominator is a perfect square and 8 has a perfect square
factor, 4:
8
4
4
2
#9 5 #9 3 2 5 #9 3 !2 5 3 !2 Answer
b. Here the numerator is a perfect square but the denominator is not. We must
multiply by 55 to write the radicand as a fraction with a denominator that is
a perfect square:
5
20
4
4
4
4
2
#5 5 #5 3 5 5 #25 5 #25 3 5 5 #25 3 !5 5 5 !5 Answer
c. Here neither the numerator nor the denominator is a perfect square. We
must first write the radicand as a fraction whose denominator is a perfect
square. Since 16 is the smallest perfect square that is a multiple of 8, we can
multiply by 22 to obtain a radicand with a denominator that is a perfect
square, 16:
3
3
6
2
#8 5 #8 3 2 5 #16 5
!6
!16
5 !6
4 Answer
Note: Part b could also be solved by an alternative method. Rewrite the radical
as the quotient of two radicals. Multiply by !5
. Then simplify:
!5
4
#5 5
5
5
!4
3 !5
!5
!5
!4 !5
!25
2 !5
5 Answer
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Simplifying Radicals
To write square root radicals with a variable in the denominator, we want
the variable in the denominator to be a perfect square, that is, to have an expo-
9a
nent that is even. For example, simplify #8b
3 if a . 0 and b . 0. To do this we
must write the radicand of the denominator as a perfect square. The smallest
perfect square that is a multiple of 8 is 8 3 2 or 16 and the smallest perfect
square that is a multiple of b3 is b3 b or b4. In order to have a denominator of
"16b4 we must multiply the given radical by #2b
2b.
9a
9a
#8b3 5 #8b3 ?
2b
2b
9
3
5 #18ab
16b4 5 #16b4 ? !2ab 5 4b2 !2ab
Alternatively, we could rewrite the radical as the quotient of radicals, mul-
tiply by "8b3, and then simplify:
3
"8b
"8b3
"72ab3
!9a
6b !2ab
5 3 !2ab
3 ?
3 5
3 2 5
8b3
4b2
"8b
"8b
" (8b )
9a
#8b3 5
Roots with Index n
The rules that we have derived for square roots are true for roots with any index.
!ab 5 !a ? !b
n
n
n
#b 5
na
and
!a
n
!b
n
n
In order to write !k in simplest form, we must find the largest factor of k
of the form an. For instance:
3
3
3
3
3
!
40 5 !
8355 !
83 !
5 5 2!
5
Recall that if the index of a root is n, then if a is a rational number, "ak is a
rational number when k is divisible by n. For example:
3 6
"
a 5 a2
4 12
"
b 5 b3
5 5
"
q 5 q1 5 q
n
n an
"
x 5 xa
In order to write "x8 in simplest form, find the largest multiple of 3 that is
less than 8. That multiple is 6. Write x8 as x6 x2.
3
3 8
3 6
3 2
3 2
"
x 5"
x ?"
x 5 x2 ? "
x
In order to simplify #q , we must write q as an equivalent fraction whose
denominator is of the form rn. For instance:
np
4 1
4 1
#2 5 #2 3
p
23
23
4 2
4
"2
"8
5#
or 12 !
8
4 4 5 2
24 5 "
2
4 3a
4 3a
#2b3c2 5 #2b3c2 ?
4
3
8bc2
8bc2
4
3
5 # 24abc
16b4c4
4
2
4
1 4
2
5#
16b4c4" 24abc
1 4
"24abc2
5 2bc
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EXAMPLE 1
Write each of the following in simplest radical form:
a. !48
3
c. "
54y8
9
b. #20b
Solution a. The factors of 48 are 2 24, 3 16, 4 12, and 6 8. The pair of factors
with the largest perfect square is 3 16:
!48 5 !16 3 3 5 !16 3 !3 5 4 !3 Answer
b. The smallest multiple of 20b that is a perfect square is 100b2. Multiply the
radicand by 5b
5b:
9
9
#20b 5 #20b ?
5b
5b
45b
5 #100b
2
9
5 #100b
2 ? 5b
9
5 #100b
2 ? !5b
3
5 10b
!5b Answer
c. The factors of 54 are 2 27, 3 18, and 6 9. The factor 27 is the cube of
3. The largest multiple of the index that is less than the exponent of y is 6:
3
3
3
3
3
"
54y8 5 "
27y6 ? 2y2 5 "
27y6 ? "
2y2 5 3y2 ? "
2y2 Answer
EXAMPLE 2
a. Write 2"50b3 in simplest radical form.
b. For what values b does 2"50b3 represent a real number?
b. 2"50b3 is a real number when
50b3 is positive or 0, that is, for
b 0. Answer
Solution a. 2"50b3 5 2"25b2 ? "2b
5 2(5b) "2b
5 10b"2b Answer
EXAMPLE 3
Write !0.2 in simplest radical form.
Solution METHOD 1
Write 0.2 as the product of a perfect square and a prime:
!0.2 5 !0.20
5 !0.04 3 5
5 !0.04 3 !5
5 0.2 !5
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Simplifying Radicals
93
METHOD 2
2
2
Write 0.2 as a common fraction, 10
. Write 10
as an equivalent fraction with a
denominator that is a perfect square, 100:
2
!0.2 5 #10
2
5 #10
3 10
10
20
5 #100
4
5 #100
3 !5
2
5 10
!5
2
Answer 0.2 !5 or 10
!5
Note: For a decimal fraction, the denominator will be a perfect square
when there are two decimal places (hundredths), four decimal places (tenthousandths) or any even number of decimal places (a denominator of 102n
for n a counting number). For example, !0.01 5 0.1, !0.0001 5 0.01,
!0.000001 5 0.001.
Exercises
Writing About Mathematics
1. Explain the difference between 2!36 and !236.
2. If a is a negative number, is 2"28a3 a positive number, a negative number, or not a real
number? Explain your answer.
3
Developing Skills
In 3–38, write each radical in simplest radical form. Variables in the radicand of an even index are
non-negative. Variables occurring in the denominator of a fraction are non-zero.
3. !12
7. "98c4
11. 12"72ab5
3
15. "
375x5y6
6
19. #81y
6
23. #3x
4y
4. !50
5. !32
3
12. !
16
3
13. !
24
8. 3"20y5
4
16. "
48a9b3
20. #a2
5
24. #5b3 5
9. 5"200xy2
17. #4a
25
4
a
21. #5b
3
25. #5a
18
6. "8b3
10. 4"363x5y7
3
14. "
40a4
18. #3b
49
3
1
22. #6xy
15
26. #8b
3
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Real Numbers and Radicals
4a
27. #27b
3
9
28. #50xy
7
35. !300c
3a
36. #
3
3
32. "2.42ab5
31. !0.5
3
29. # 8x3
5y
33. "0.001x3
4 2a
37. #
b2c3
8
30. !0.08
34. "128x2
4
38. "
32x5y4
Applying Skills
39. The lengths of the legs of a right triangle are 8 centimeters and 12 centimeters. Express the
length of the hypotenuse in simplest radical form.
40. The length of one leg of an isosceles right triangle is 6 inches. Express the length of the
hypotenuse in simplest radical form.
41. The length of the hypotenuse of a right triangle is 24 meters and the length of one leg is 12
meters. Express the length of the other leg in simplest radical form.
42. The dimensions of a rectangle are 15 feet by 10 feet. Express the length of the diagonal in
simplest radical form.
43. The area of a square is 150 square feet. Express the length of a side of the square in simplest radical form.
44. The area of a circular pool is 5x2y4p square meters. Express the radius of the pool in simplest radical form.
5
5 5
45. The area of a triangle is "
243x5y10 square units. If the length of a side of the triangle is "
x,
express the length of the altitude to that side in simplest radical form.
46. Tom has a trunk that is 18 inches wide, 32 inches long, and 16 inches high. Which of the following objects could Tom store in the trunk? (There may be more than one answer.)
(1) A walking stick 42 inches long
(2) A fishing rod 37 inches long
(3) A yardstick
(4) A baseball bat 34 inches long
3-4 ADDING AND SUBTRACTING RADICALS
Recall that the sum or difference of fractions or of similar algebraic terms is
found by using the distributive property.
2
5
1 45 5 15 3 2 1 15 3 4
5 15 (2 1 4)
5
5
1
5 (6)
6
5
2x 1 4x 5 x(2 1 4)
5 x(6)
5 6x
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95
We do not write all of the steps shown above when adding fractions or algebraic terms. The principles that justify each of these steps assure us that the
results are correct.
We can apply the same principles to the addition or subtraction of radical
expressions. To express the sum or difference of two radicals as a single radical,
the radicals must have the same index and the same radicand, that is, they must
be like radicals. We can use the same procedure that we use to add or subtract
like terms.
2 !5 1 4 !5
4 !3b 2 3 !3b
3
3
!
2 1 7!
2
5 !5(2 1 4)
5 !3b(4 2 3)
3
5 !
2(1 1 7)
5 6 !5
5 !3b
3
5 8!
2
5 !5(6)
5 !3b(1)
3
5 !
2(8)
Two radicals that do not have the same radicand or do not have the same
index are unlike radicals. The sum or difference of two unlike radicals cannot be
expressed as a single radical if they cannot be written as equivalent like radicals.
3
4
Each of the following, !2 1 !3, ! 2 1 !2, and !2 1 !3, are the sums of
unlike radicals and are in simplest radical form.
Simplifying Unlike Radicals
The fractions 31 and 14 do not have common denominators but we can add these
two fractions by writing them as equivalent fractions with a common denominator. The radicals !8 and !50 have the same index but do not have the same
radicand. We can add these radicals if they can be written in simplest form with
a common radicand.
!8 5 !4!2 5 2 !2
and
!50 5 !25!2 5 5 !2
"12b3 5 "4b2 !3b 5 2b !3b
and
"27b3 5 "9b2 !3b 5 3b !3b
!8 1 !50 5 2 !2 1 5 !2 5 7 !2
"12b3 1 "27b3 5 2b !3b 1 3b !3b 5 5b !3b
Note that it is not always possible to express two unlike radicals as like radicals.
For example, the sum !2 1 !3 cannot be written with the same radicand and
therefore the sum cannot be written as one radical.
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EXAMPLE 1
Write !12 1 !20 2 !45 1 #13 in simplest form.
Solution Simplify each radical and combine like radicals.
!12 1 !20 2 !45 1 #13 5 !4!3 1 !4!5 2 !9!5 1 #39
5 2 !3 1 2 !5 2 3 !5 1 13 !3
5 2 !3 1 13 !3 1 2 !5 2 3 !5
5 73 !3 2 !5 Answer
EXAMPLE 2
1
Simplify: 3x#3x
1 !300x
Solution Simplify each radical.
1
!3x
!3x
3x
1
1
3x#3x
5 3x#3x
? 3x
3x 5 3x# 2 5 3x 3x 5 3x 3x 5 !3x
9x
!300x 5 !100!3x 5 10 !3x
1
Add the simplified radicals.
1
3x#3x
1 !300x 5 !3x 1 10 !3x 5 11 !3x (x . 0) Answer
EXAMPLE 3
Solve and check: 4x 2 !8 5 !72
Solution
4x 2 !8 5 !72
4x 2 !8 1 !8 5 !72 1 !8
4x 5 !36A !2 B 1 !4A !2 B
4x 5 6 !2 1 2 !2
4x 5 8 !2
4x
4
Answer x 5 2 !2
5
8 !2
4
x 5 2 !2
Check
4x 2 !8 5 !72
4(2!2) 2 !4A !2B 5 !36A !2B
?
8 !2 2 2 !2 5 6 !2
?
6 !2 5 6 !2 ✔
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97
Exercises
Writing About Mathematics
1
1
1. Danielle said that 3x#3x
could be simplified by writing 3x#3x
as #9x
3x 5 !3x. Do you
2
agree with Danielle? Justify your answer.
2. Does !16 1 !48 5 !64? Justify your answer.
Developing Skills
In 3–38 write each expression in simplest form. Variables in the radicand with an even index are
non-negative. Variables occurring in the denominator of a fraction are non-zero.
3. !2 1 5 !2
4. 6 !5 2 4 !5
7. !50 1 !2
8. 3 !5y 2 !20y
5. 8 !3 1 !3
9. "250a2 1 "10a2
11. "24xy2 1 "54xy2
13. "98c5 2 "18c5
15. 4b"24b3 1 "54b5
17. !5 1 #15
19. 14#17 1 !28
21. a !45 1 "20a2 2 5 !2a
23. 2 !3y 2 5y2 1 4 !3y 1 "36y4
25. !12 2 !24 1 !48 1 !27
27. #16 1 #83 2 #23
3
3
29. !54 1 !128
31. !9x 1 !25x
33. !8a 2 !2a
35. "63a2 2 "45a2
37. "50x3 1 "200x3
6. 5 !7 2 !7
10. 8"11b4 2 "99b4
12. "200a7 2 "50a7
14. x !32x 1 "128x3
16. 3x3 !80 1 2"125x6
18. !24 1 2#32
1
1
1 #2x
20. #2x
22. x !600 2 2"24x2 1 4x !96
24. "162a4b3 1 3 2 ab"18a2b 2 1
1
1 !20
26. 5#15 2 #10
3
3
28. !2 1 !16
4
4
30. !48 2 !3
32. !100y 2 !25y
34. "18b2 1 "800b2
36. "4ab2 2 "ab2
38. "49x3 2 2x !4x
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Real Numbers and Radicals
In 39–42, solve and check each equation.
39. 5x 2 !3 5 !48
41. 4a 1 !6 5 a 1 !96
40. 12y 1 !32 5 !200
42. y 1 !20 5 !45 2 2y
Applying Skills
In 43–47, express each answer in simplest radical form.
43. The lengths of the sides of a triangle are !75 inches, !27 inches, and !108 inches. What is
the perimeter of the triangle?
44. The length of each of the two congruent sides of an isosceles triangle is !500 feet and the
length of the third side is !45 feet. What is the perimeter of the triangle?
45. The lengths of the legs of a right triangle are !18 centimeters and !32 centimeters.
a. Find the length of the hypotenuse.
b. Find the perimeter of the triangle.
46. The length of each leg of an isosceles right triangle is !98 inches.
a. Find the length of the hypotenuse.
b. What is the perimeter of the triangle?
47. The dimensions of a rectangle are !250 meters and !1,440 meters.
a. Express the perimeter of the rectangle in simplest radical form.
b. Express the length of the diagonal of the rectangle in simplest radical form.
3-5 MULTIPLYING RADICALS
Recall that if a and b are non-negative numbers, !ab 5 !a ? !b. Therefore,
by the symmetric property of equality, we can say that !a ? !b 5 !ab. Recall
also that for any positive number a, !a ? !a 5 "a2 5 a. We can use these
rules to multiply radicals.
For example:
!4 ? !25 5 !100 5 10
!2 ? !2 5 !4 5 2
!8 ? !2 5 A !4 ? !2B ? !2 5 2A !2 ? !2B 5 2(2) 5 4
"6a3 ? !18a 5 "108a4 5 "36a4 ? !3 5 6a2 !3 (a 0)
Note: !22 3 !28 2 !16 because !22 and !28 are not real numbers.
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99
The rule for the multiplication of radicals is true for any two radicals with
n
n
n
the same index if the radicals are real numbers. That is, since !ab 5 !a ? !b
is true for non-negative real numbers, the following is also true by the symmetric property of equality:
n
n
n
!
a? !
b5 !
ab (a, b 0)
Note: This rule does not apply when the index n is even and a or b are nega4
4
4
4
4
tive. For instance, !22 3 !28 2 !16 because !22 and !28 are not real
numbers.
EXAMPLE 1
Answers
Simplify:
a. !8 ? !27
3
3
5 !
216 5 6
3
or
52356
4
4 x
b. "
48x2 ? #
3
2
4
4
5#
48x2 ? x3 5 "
16x4 5 2x (x 0)
2
Multiplying Sums That Contain Radicals
The distributive property for multiplication over addition or subtraction is true
for all real numbers. Therefore, we can apply it to irrational numbers that contain radicals.
For example:
!3A2 1 !3B 5 !3(2) 1 !3A !3B 5 2 !3 1 3
A2 1 !5B A1 1 !5B 5 2A1 1 !5B 1 !5A1 1 !5B
5 2 1 2 !5 1 !5 1 5 5 7 1 3 !5
A3 1 !2B A3 2 !2B 5 3A3 2 !2B 1 !2A3 2 !2B
5 9 2 3!2 1 3 !2 2 2 5 7
In each of these examples, we are multiplying irrational numbers. We know
that !3, !5 and !2 are all irrational numbers. The sum of a rational number
and an irrational number is always an irrational number. Therefore, 2 1 !3,
2 1 !5, 1 1 !5, 3 1 !2, and 3 2 !2 are all irrational numbers. The product
of irrational numbers may be a rational or an irrational number.
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EXAMPLE 2
Express each of the following products in simplest form:
a. !5A !10B
b. A3 1 !6aB A1 1 !2aB
Solution a. !5A !10B 5 !50 5 !25 ? !2 5 5 !2 Answer
c. A !5 1 !7BA !5 2 !7B
b. A3 1 !6aB A1 1 !2aB 5 3A1 1 !2aB 1 !6aA1 1 !2aB
5 3 1 3 !2a 1 !6a 1 "12a2
5 3 1 3!2a 1 !6a 1 "4a2 A !3B
5 3 1 3!2a 1 !6a 1 2a !3 Answer
c. A !5 1 !7B A !5 2 !7B 5 !5A !5 2 !7B 1 !7A !5 2 !7B
5 5 2 !35 1 !35 2 7
5 22 Answer
EXAMPLE 3
The length of a side, s, of an equilateral triangle is !5 inches and the length of
the altitude, h, is #15
4 inches. Find the area of the triangle.
Solution
Area of a triangle 5 12bh
5 12 A !5B Q #15
4 R
5 12 Q #75
4 R
5 12 Q #25
4 ? !3 R
5 12 A 52 !3 B
5 54 !3 sq inches Answer
Exercises
Writing About Mathematics
1. Brandon said that if a is a positive real number, then 3a, 4a, and 5a are the lengths of the
sides of a right triangle. Therefore, 3 !2, 4 !2, and 5 !2 are the lengths of the sides of a right
triangle. Do you agree with Brandon? Justify your answer.
4
2. Jennifer said that if a is a positive real number, then "a2 5 !a. Do you agree with
Jennifer? Justify your answer.
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101
Developing Skills
In 3–41, express each product in simplest form. Variables in the radicand with an even index are nonnegative.
3. !2 ? !8
6. !8 ? !12
9.
1
#3
? !24
12. A !12B 2
15. "x3 ? !4x
18. "x5y3 ? !3xy
21. #a3 ? #a4
2
4
4
24. !
27 ? !
3
27. !8A6 1 !2B
30. A1 1 !5B A3 2 !5B
33. A7 1 !5bB A7 2 !5bB
36. A !6 1 6cB A !6 2 6cB
39. A3 1 "5ab3 B 2
4. !5 ? !45
7. 2!10 ? !18
10. !21 ?
13. A3 !3B 2
4
#3
16. 2 !ab ? 2"ab2
19. 7 !a ? 5#a9
3
3
22. !
2 ? !4
25. !2 A2 1 !2B
28. !5aA !5a 2 3B
31. A9 1 !2bB A1 1 !2bB
4
4
34. Ax 2 !
3y BA2x 2 !
3y B
37. Aa 1 !bB Aa 2 !bB
40. A1 2 !7B A1 1 !7B A1 1 !7B
5. !3 ? !27
8. 3 !2 ? !10
5
11. 8 !6 ? #12
14. A22 !5B 2
17. !5y ? "4y3
20. #x2 ? #x2
2
3
3
23. "
15a2 ? "9a4
26. !5A1 2 !10B
29. "12xy3 A !3xy 1 3B
32. A7 1 !5yB A3 2 !5yB
35. A !6 1 6BA !6 2 7B
38. A1 2 !3B 2
41. A2 2 !5BA2 1 !5B 2
Applying Skills
42. The length of a side of a square is 48 !2 meters. Express the area of the square in simplest
form.
43. The dimensions of a rectangle are 12 !2 feet by !50 feet. Express the area of the rectangle
in simplest form.
44. The dimensions of a rectangular solid are !5 inches by A2 1 !3B inches by A2 2 !3B
inches. Express the volume of the solid in simplest form.
45. The lengths of the legs of a right triangle in feet are A3 1 !3B and A3 2 !3B .
a. Find the length of the hypotenuse of the triangle.
b. Express, in simplest form, the perimeter of the triangle.
c. What is the area of the triangle?
46. The radius of the surface of a circular pool is A 2 1 "xy5 B meters. Express the area of the
pool in simplest form.
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3-6 DIVIDING RADICALS
We can use what we know about the relationship between division and multiplication and the rule for the multiplication of radicals to write a rule for the
division of radicals. We know that for any non-negative a and positive b,
!a
!a 4 !b 5 !b
. Therefore, if a and b are positive real numbers, then:
!a
5 #ba
!a 4 !b 5 !a ? #b1 5 #ba or !b
Recall that a similar rule is true for roots with any index, and so we can
write:
n
n
n a
a
!
a4 !
b5 !
5#
n
b
!b
n
3
3 28
3 28 3
3
! 28
2
Note: #
5 22
27 5 3
3 5 23 is a true statement because #27 , !28, and ! 27
! 27
!24
24
are all real numbers. However, we cannot write #24
9 5 !9 because # 9 and
!24 are not real numbers.
EXAMPLE 1
Express !27a 4 "3a3 in simplest radical form when a is a positive real
number.
9
3a
3
9
!27a 4 "3a3 5 !27a3 5 #27a
3a3 5 Ä a2 ? 3a 5 #a2 5 a Answer
"3a
1
1
Solution
!9( !3a)
!27a 4 "3a3 5 !27a3 5
5 a3 !3a
5 a3 Answer
2
!3a
"3a
"a ( !3a)
1
Alternative
Solution
1
EXAMPLE 2
!15
Write !45
in simplest radical form.
Solution
Alternative
Solution
!15
!45
!3
3
3
1
1
5 #15
45 5 #3 5 #3 3 3 5 #9 5 3 Answer
!15
!45
3 !5
!3
5 !3
5 !3
3 3 1 5 3 Answer
!9 3 !5
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Dividing Radicals
103
EXAMPLE 3
1 8 !15b
Simplify: 12 !10a
4 !5ab
Solution Use the distributive property of division over addition.
12 !10a 1 8 !15b
4 !5ab
!10a
!15b
5 12
1 48 !5ab
4 !5ab
5 3#b2 1 2#a3
5 3#b2 ? bb 1 2#34 ? aa
3a
5 3#2b
b2 1 2# a2
5 b3 !2b 1 a2 !3a (a . 0, b . 0) Answer
Exercises
Writing About Mathematics
1. Jonathan said that !10
2 5 !5. Do you agree with Jonathan? Justify your answer.
2. Show that the quotient of two irrational numbers can be either rational or irrational.
Developing Skills
In 3–29 write each quotient in simplest form. Variables in the radicand with an even index are
non-negative. Variables occurring in the denominator of a fraction are non-zero.
3. !24 4 !6
6. "50a3 4 !5a
9. !54
!2
12.
"80x2y
"30xy2
4. !75 4 !3
7. "24x2 4 "3x3
10. !300
!25
13. !27b2
"6b
7
15. !7y
12a2
16. "!4a
2 !5
21. !20!5
1 !3
22. !48!3
18. 4 !2 21!28 !12
6 !5
24. 5 1!5
27.
4
"c6
"c
5
5
9 !50
19. 3 !103 2
!5
3
3
3
5
25. "27x 3 1 3"36x
"3x
2
28. "324w4
3
"3w
5. !72 4 !8
8. !150
!3
35a3
11. "
!10a
3
14. !3x
18c3
17. "!9c
1 !54
20. !72!18
1 !15
23. !10!10
4
ab4
26. "
4 2 4
"a b
4
4
6
29. "64x 41 6"40x
"x
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Applying Skills
30. The area of a rectangle is 25 !35 square feet and the width is 10 !5 feet. Find the length of
the rectangle in simplest radical form.
31. The area of a right triangle is 6 !2 square centimeters and the length of one leg is !12 centimeters.
a. What is the length of the other leg?
b. What is the length of the hypotenuse?
3-7 RATIONALIZING A DENOMINATOR
To rationalize the denominator of a fraction means to write the fraction as an
equivalent fraction with a denominator that is a rational number.
A fraction is in simplest form when the denominator is a positive integer,
that is, a rational number. For example, we found the simplest form of a fraction
such as !5
by multiplying the numerator and denominator by !2 so that the
!2
denominator would be an integer.
5
5
10
2
#2 5 #2 3 2 5 # 4 5
!10
!4
1
5 !10
2 5 2 !10
How do we write the fraction 2 13 !6 in simplest form, that is, with a rational denominator? To do so, we must multiply numerator and denominator of
the fraction by some numerical expression that will make the denominator a
rational number.
Recall that (a + b)(a 2 b) 5 a2 2 b2. For example:
A1 1 !2B A1 2 !2B
5 1 2 A !2B
2
5122
5 21
2
A5 2 !3B A5 1 !3B
5 5 2 A !3B
2
2
A2 1 !6B A2 2 !6B
5 22 2 A !6B 2
5 25 2 3
5426
5 22
5 22
The binomials (a + b) and (a 2 b) are called conjugates. We can use this last
!6
example to simplify 2 13 !6. If we multiply the fraction by 1 in the form 22 2
,
2 !6
we will have a fraction whose denominator is an integer.
3
2 1 !6
!6
6 2 3 !6
5 2 13 !6 3 22 2
5 62 2 3 !6 2 5 6 4223 !6
5 26 12 3 !6
22
6 5
2 !6
2 2 A !6B
If we use a calculator to find a rational approximation for the given fraction
and for the simplest form of the given fraction, these approximations will be
equal.
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Rationalizing a Denominator
ENTER: 3
6
DISPLAY:
(
)
)
2 1 2nd
¯
ENTER:
6
ENTER
3/(2+√(6))
.6742346142
26 3 2nd
(
DISPLAY:
)
)
105
¯
2 ENTER
( -6 + 3 √ ( 6 ) ) / 2
.6742346142
Recall that when the given numerical expression is written as a fraction, the
line of the fraction is a grouping symbol. When entering these fractions into the
calculator, the numerator or denominator that is written as a sum or a difference
must be enclosed by parentheses.
To write the fraction 2a 14 !8b with a rational denominator, multiply the
2 !8b
fraction by 2a
and simplify the result.
2a 2 !8b
4
2a 1 !8b
4A2a 2 !8bB
2 !8b
? 2a
5 4a2 2 8b
2a 2 !8b
4A2a 2 2 !2bB
5 4(a2 2 2b)
8Aa 2 !2bB
5 4(a2 2 2b)
5
The fraction
2Aa 2 !2bB
a2 2 2b
2Aa 2 !2bB
a2 2 2b
has a rational denominator if a and b are rational
numbers.
EXAMPLE 1
7
Write 2 !7
as an equivalent fraction with a rational denominator.
Solution When !7 is multiplied by itself, the product is 7, a rational number. Multiply
!7
the fraction by 1 in the form !7
.
!7
!7
7
5 2 !7
3 !7
5 72(7)
5 72(7)
5 !7
2 Answer
!7
1
7
2 !7
1
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Real Numbers and Radicals
EXAMPLE 2
!2
Rationalize the denominator of the fraction 33 1
.
2 !2
Solution The conjugate of 3 2 !2 is 3 1 !2. Multiply the fraction by 1 in the form
3 1 !2
.
3 1 !2
3 1 !2
3 2 !2
!2
!2
3 !2 1 2
5 33 1
3 33 1
5 9 1 3 !29 1
5 11 17 6 !2 Answer
2 !2
1 !2
2 2
Note: The numerator is A3 1 !2B 2 or 32 1 2 A3 !2B 1 A !2B 2.
EXAMPLE 3
6
Find the sum: 3 2!2!6 1 !8
Solution Rationalize the denominator of each fraction.
!2
3 2 !6
6
!8
3 !2 1 !4( !3)
!6
1 !12
2 !3
5 3 2!2!6 3 33 1
5 3 !2
5 3 !2 1
2
2 5
3
9 2 6
1 !6
3 2 A !6B
!2
3 !2
6
5 !8
3 !2
5 6!16
5 6 !2
4 5 2
!2
Add the fractions. In order to add fractions, we need a common denominator.
2 !3
The common denominator of 3 !2 1
and 3 !2
3
2 is 6.
!2
3 2 !6
2 !3
6
3
1 !8
5 3 !2 1
3 22 1 3 !2
3
2 3 3
4 !3
5 6 !2 1
1 9 !2
6
6
1 9 !2
5 6 !2 1 4 !3
6
5 15 !2 61 4 !3 Answer
EXAMPLE 4
Solve for x and check the solution: x !2 5 3 2 x
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Rationalizing a Denominator
How to Proceed
Solution
107
x !2 5 3 2 x
(1) Write an equivalent equation with
only terms in x on the left side of
the equation and only constant
terms on the right side. Add x to
each side of the equation:
x !2 1 x 5 3 2 x 1 x
x !2 1 x 5 3
xA !2 1 1B 5 3
(2) Factor the left side of the equation:
(3) Divide both sides of the equation by
the coefficient of x:
xA !2 1 1B
!2 1 1
3
5 !2 1 1
2 1
x 5 !2 31 1 3 !2
!2 2 1
(4) Rationalize the denominator:
x5
x5
3A !2 2 1B
2 2 1
3A !2 2 1B
1
x 5 3 !2 2 3
(5) Check:
x !2 5 3 2 x
A3 !2 2 3B A !2B 5 3 2 A3 !2 2 3B
?
3(2) 2 3 !2 5 3 2 3 !2 1 3
?
6 2 3 !2 5 6 2 3 !2 ✔
Exercises
Writing About Mathematics
7
1. Justin simplified 2 !7
by first writing 7 as !49 and then dividing numerator and
denominator by !7.
a. Show that Justin’s solution is correct.
7
b. Can 2 !5
be simplified by using the same procedure? Explain why or why not.
!8
2. To rationalize the denominator of 2 14 !8, Brittany multiplied by 22 2
and Justin
2 !8
!2
multiplied by 11 2
. Explain why both are correct.
2 !2
Developing Skills
In 3–38, rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
1
3. !3
7. 515
!3
5
4. !10
4
8. 8 !6
4
5. !2
12
9. !27
4
6. 2 !3
6
10. !12
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8
11. !24
12. 24 !2
!3
5 !5
13. 15
!2
14. 2!24
!6
15. 3 11 !5
16. 5 21 !2
17. 1 11 !3
18. 3 24 !3
19. 1 13 !5
20. 4 14 !7
21. !5 32 2
22. !7 91 2
23. 2 2!2!2
!2
27. 23 1
2 !2
24. 3 16 !3
2 !7
28. 46 2
!7
2 !x 2 5 !y
2
31. ba21!2
32. !x 1 !y
2
2
1 !y
35. !x
2
36. !x 11 6 2 !6
!20y
!5x
25. !5x
2 2
26. y !5 1 1
2 1
29. !10
!10 1 1
!5
30. 77 1
2 !5
33. !z 41 8
!a
!b
1 2!a
37. 2!b
34. !a!a
2 2
5
38. x 13 !2 1 !x
In 39–42: a. Write each fraction in simplest radical form. b. Use a calculator to find a rational
approximation for the given fraction. c. Use a calculator to find a rational approximation for the
fraction in simplest form.
40. 2 1 "3
39. 4
"6
41.
"3
4
"3 2 1
42.
3 1 "7
3 2 "7
In 43–46, solve and check each equation.
43. 2a 1 "50 5 "98
45. y"3 1 1 5 3 2 y
44. 5x 2 "12 5 "108 2 3x
46. 7 2 b"8 5 b"5 1 4
Applying Skills
47. The area of a rectangle is 24 square inches. The length of the rectangle is !5 1 1 inches.
Express the width of the rectangle in simplest form.
48. The perimeter of an isosceles triangle is !50 feet. The lengths of the sides are in the ratio
3 : 3 : 4. Find the length of each side of the triangle.
3-8 SOLVING RADICAL EQUATIONS
An equation that contains at least one radical term with a variable in the radicand is called a radical equation. For example, !2x 2 3 5 5 is a radical equation. Since the radical is a square root, we can solve this equation by squaring
both sides of the equation.
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109
Solving Radical Equations
Solution:
!2x 2 3 5 5
A !2x 2 3B 5 5
2
2
2x 2 3 5 25
2x 5 28
Check:
!2x 2 3 5 5
!2(14) 2 3 5 5
?
!28 2 3 5 5
?
!25 5 5
?
555✔
x 5 14
Note that squaring both sides of a radical equation does not always result in
an equivalent equation. For example, if the given equation had been
!2x 2 3 5 25, the equation obtained by squaring both sides would have been
2x 2 3 5 25, the same as the equation obtained by squaring both sides of
!2x 2 3 5 5. The solution of 2x 2 3 5 25 is not a root of !2x 2 3 5 25. The
radical !2x 2 3 represents the principal root of 2x 2 3, which is a positive
number. Therefore, the equation !2x 2 3 5 25 has no real root. There is no
real number such that !2x 2 3 5 25.
Often an equation must be rewritten as an equivalent equation with the radical alone on one side of the equation before squaring both sides of the equation. For example, to solve the equation x 2 !5 2 x 5 3, we must first add 2x
to both sides of the equation to isolate the radical.
x 2 !5 2 x 5 3
Check: x 5 4
2!5 2 x 5 3 2 x
A2!5 2 xB 2 5 (3 2 x) 2
5 2 x 5 9 2 6x 1 x2
0 5 4 2 5x 1 x2
0 5 (4 2 x)(1 2 x)
Check: x 5 1
x 2 !5 2 x 5 3
x 2 !5 2 x 5 3
4 2 !1 5 3
1 2 !4 5 3
4 2 !5 2 4 5 3
?
?
?
4 2 153
353✔
1 2 !5 2 1 5 3
?
?
?
1 2 253
21 2 3 ✘
0542x 0512x
x54
x51
The root of this equation is x 5 4.
Note that after squaring both sides of the equation, we have a quadratic
equation that has two real roots. One of these is the root of the given equation,
x 2 !5 2 x 5 3 which is equivalent to 2!5 2 x 5 3 2 x. The other is the
root of the equation 2!5 2 x 5 x 2 3. When we square both sides of either
of these equations, we obtain the same equation, 5 2 x 5 9 2 6x 1 x2.
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EXAMPLE 1
Solve and check: 4 1 !1 2 3x 5 12
Solution
How to Proceed
(1) Isolate the radical by adding 24
to both sides of the equation:
4 1 !1 2 3x 5 12
!1 2 3x 5 8
A !1 2 3xB 2 5 82
(2) Square both sides of the equation:
1 2 3x 5 64
(3) Add 21 to both sides of the equation:
23x 5 63
x 5 221
(4) Divide both sides of the equation by 23:
(5) Check:
4 1 !1 2 3x 5 12
4 1 !1 2 3(221) 5 12
?
4 1 !1 1 63 5 12
?
4 1 !64 5 12
?
?
4 1 8 5 12
12 5 12 ✔
Answer x 5 221
EXAMPLE 2
3
Find the value of a such that !4 2 2a 5 22.
Solution
How to Proceed
(1) The radical is already isolated. Since the
radical is a cube root, cube both sides of
the equation:
(2) Solve the resulting equation for a:
3
!
4 2 2a 5 22
3
A !4 2 2a B 3 5 (22) 3
4 2 2a 5 28
22a 5 212
a56
(3) Check:
!3 4 2 2a 5 22
3
!
4 2 2(6) 5 22
?
3
!
4 2 12 5 22
?
3
!
28 5 22
22 5 22 ✔
?
Answer a 5 6
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Solving Radical Equations
111
EXAMPLE 3
What is the solution set of x 5 1 1 !15 2 7x?
Solution
How to Proceed
x 5 1 1 !15 2 7x
(1) Write the equation with the
radical alone on one side of the
equation by adding 21 to both
sides of the equation:
x 2 1 5 !15 2 7x
(x 2 1) 2 5 A !15 2 7xB 2
x 2 2x 1 1 5 15 2 7x
(2) Square both sides of the equation:
2
x2 1 5x 2 14 5 0
(3) Write the quadratic equation in
standard form:
(x 1 7)(x 2 2) 5 0
(4) Factor the polynomial.
(5) Set each factor equal to 0 and
solve for x:
x1750
x 5 27
x2250
x52
(6) Check:
Let x 5 27:
x 5 1 1 !15 2 7x
27 5 1 1 !15 2 7(27)
?
27 5 1 1 !15 1 49
?
27 5 1 1 !64
?
?
Let x 5 2:
x 5 1 1 !15 2 7x
2 5 1 1 !15 2 7(2)
?
2 5 1 1 !15 2 14
?
2 5 1 1 !1
?
?
27 5 1 1 8
251 1 1
27 2 9 ✘
252✔
27 is not a solution of x 5 1 1 !15 2 7x and 2 is a solution of
x 5 1 1 !15 2 7x.
Answer The solution set of x 5 1 1 !15 2 7x is {2}.
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EXAMPLE 4
Solve the equation !x 1 5 5 1 1 !x.
How to Proceed
(1) Square both sides of the equation. The
right side will be a trinomial that has
a radical in one of its terms:
A !x 1 5B 2 5 A1 1 !xB 2
x 1 5 5 1 1 2 !x 1 x
(2) Simplify the equation obtained in step 1
and isolate the radical:
(3) Square both sides of the equation
obtained in step 2:
42 5 A2 !xB 2
16 5 4x
x54
(4) Solve for the variable:
(5) Check:
4 5 2!x
!x 1 5 5 1 1 !x
!4 1 5 5 1 1 !4
?
353✔
Answer x 5 4
Exercises
Writing About Mathematics
1. Explain why !x 1 3 , 0 has no solution in the set of real numbers while !x 1 3 $ 0 is
true for all real numbers greater than or equal to 23.
2. In Example 3, are x 2 1 5 !15 2 7x and (x 2 1) 2 5 A !15 2 7xB 2 equivalent equations?
Explain why or why not.
Developing Skills
In 3–38, solve each equation for the variable, check, and write the solution set.
3. !a 5 5
6. !4y 5 12
9. 1 1 !x 5 3
12. !5 1 a 5 7
15. 3 2 !2x 1 5 5 0
18. !20 2 2x 5 !5x 2 8
4. !x 5 7
7. 2 !b 5 8
5. 4 !y 5 12
8. !2b 5 8
10. !1 1 x 5 3
11. 5 1 !a 5 7
16. 8 1 !2x 2 1 5 15
17. !5x 1 2 5 !9x 2 14
13. 3 2 !y 5 1
19. x 5 !4x 1 5
14. !3 2 y 5 1
20. y 5 !y 1 2
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Chapter Summary
21. x 1 3 5 !1 2 3x
22. a 2 2 5 !2a 2 1
24. 3 2 b 5 !3b 2 11
25. x 5 1 1 !x 1 11
30. !x 5 2
31. 5 1 !a 1 2 5 3
27. x 1 4 !x 5 5
3
28. 3 1 !4x 2 3 5 2x
3
3
33. 1 2 !x 1 7 5 4
36. 1 1 !2x 5 !3x 1 1
4
34. 210 1 !n 2 2 5 28
37. !x 1 4 2 !x 2 1 5 1
113
23. b 2 3 5 !3b 2 11
26. x 1 !x 1 1 5 5
29. 2 1 !3x 2 2 5 3x
3
32. 2 1 !3b 2 2 5 6
35. !x 1 5 5 1 1 !x
38. !5x 2 !x 1 4 5 2
Applying Skills
39. The lengths of the legs of an isosceles right triangle are x and !6x 1 16. What are the
lengths of the legs of the triangle?
40. The width of a rectangle is x and the length is !x 2 1. If the width is twice the length, what
are the dimensions of the rectangle?
41. In ABC, AB 5 !7x 1 5, BC 5 !5x 1 15, and AC 5 !2x.
a. If AB = BC, find the length of each side of the triangle.
b. Express the perimeter of the triangle in simplest radical form.
CHAPTER SUMMARY
An irrational number is an infinite decimal number that does not repeat.
The union of the set of rational numbers and the set of irrational numbers is the
set of real numbers.
If the domain is the set of real numbers, the elements of the solution set of
an inequality usually cannot be listed but can be shown on the number line.
If a 0 and x 5 a, then x 5 a or x 5 2a. These two points on the real number line separate the line into three segments. The segment between a and 2a is
the graph of the solution set of x , a and the other two segments are the graph
of the solution set of x . a.
A square root of a positive number is one of the two equal factors whose
product is that number. The principal square root of a positive real number is its
positive square root. The cube root of k is one of the three equal factors whose
product is k. The nth root of k is one of the n equal factors whose product is k.
n
The nth root of k is written as !k where k is the radicand, n is the index, and
n
!
k is the radical.
n
An irrational number of the form ! a is in simplest form when a is an integer that has no integral factor of the form bn.
To express the sum or difference of two radicals as a single radical, the radicals must have the same index and the same radicand, that is, they must be like
radicals. The distributive property is used to add like radicals.
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Real Numbers and Radicals
For any non-negative real numbers a and b:
!a ? b 5 !a ? !b
!a
5 #ba (b 0)
!a 4 !b 5 !b
!a ? !a 5 "a2 5 a
For any non-negative real number a and natural number n:
"a2n 5 an
n
n
If !a and !b are real numbers, then:
n
n
n
!
a ?b5 !
a? !
b
n
n
n a
a
!
a4 !
b5 !
5#
n
b (b 0)
!b
n
To rationalize the denominator of a fraction means to write the fraction as
an equivalent fraction with a denominator that is a rational number.
The binomials a + b and a 2 b are conjugate binomials. When the denominator of a fraction is of the form a 6 !b or !a 6 !b, we can rationalize the
fraction by multiplying by a fraction whose numerator and denominator are the
conjugates of the given denominator.
An equation that contains at least one radical term with a variable in the
radicand is called a radical equation. To solve an equation that contains a square
root radical, isolate the radical and square both sides of the equation. The
derived equation may or may not be equivalent to the given equation and a
check is necessary.
VOCABULARY
3-1 Irrational numbers • Real numbers • Interval notation
3-2 Square root • Principal square root • Cube root • nth root • Radicand •
Index • Radical • Principal nth root
3-3 Simplest form
3-4 Like radicals • Unlike radicals
3-7 Rationalize the denominator • Conjugates
3-8 Radical equation
REVIEW EXERCISES
In 1–8, find and graph the solution set of each inequality.
1. 1 2 4x , 2
u
u
3. x 2 12 # 11
2. 2x 1 3 # 8
4. 24x 1 5 . 13
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Review Exercises
u
u
6
.1
6. x 1
7
5. 4 2 5x $ 7
u
115
u
7. 5x 42 3 # 2
8. x 2 1 1 2 , 3
In 9–41, write each expression in simplest radical form. Variables representing
indexes are positive integers. Variables in the radicand of an even index are nonnegative. Variables occurring in the denominator of a fraction are non-zero.
9. !128
10. #34
12. !45 2 !20
15. !24 1 !20 1 !54
18. !2A3 1 !8B
21. A1 1 !2B A1 2 !2B
24. !24
!12
27. 4 2!3!3
33. "256x4 2 "36x4
4
4
64a5
36. "!4a
41. "x
n
n12
16. !8 3 !18
19. !40A !10 2 !20B
22. A5 2 !3B A5 1 !3B
!5
25. 10 1
!5
28. !9a 1 !25a
4
30. "12x5
!a
39. 44 2
1 !a
13. 3 !12 1 5 !27
31. #ba
34. "x3y4 ? "xy
3
7
37. "327b
!3b
n n14
40. "an 4
? "x
n
11. !75 1 !300
14. 12 !72 1 13 !18
17. 3 !5 3 !125
20. 6 !2A1 1 !50B
23. A2 1 !3B A1 1 !3B
26. !8141 1
29. "8b3 1 "50b3
32. 16"x5 2 "16x5
35. "18a5 ? "3a4
3
3
38. 3 21 !x
"a
n22
, where n is an integer . 2
In 42–49, solve each equation and check.
42. !2x 2 1 5 5
43. !3x 2 1 5 !2x 1 4
46. y 5 !7y 2 12
47. b 5 3 1 !29 2 5b
44. 3 1 !x 1 7 5 8
48. x 1 1 1 !6 2 2x 5 0
45. 8 1 !x 2 5 5 11
49. "a2 2 2 5 !a 1 4
50. The lengths of the sides of a triangle are 8 feet, !75 feet, and A4 1 !12B
feet. Express the perimeter of the triangle is simplest radical form.
51. The length of each leg of an isosceles right triangle is 2 !2 meters.
a. Find the length of the hypotenuse of the triangle in simplest radical form.
b. Find the perimeter of the triangle in simplest radical form.
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52. Show that x, !2x 1 1, and x 1 1 can be the lengths of the sides of a right
triangle for all x . 0.
53. The area of a rectangle is 12 square feet and the length is 2 1 !3 feet.
a. Find the width of the rectangle is simplest radical form.
b. Find the perimeter of the rectangle in simplest radical form.
Exploration
As mentioned in the chapter opener, Euclid proved that !2 is irrational.
Following is a proof by contradiction based on Euclid’s reasoning.
Assume that !2 is a rational number. Then we can write !2 5 ba with a and
b integers and b 0. Also assume that ba is a fraction in simplest form. Therefore:
2
2 5 ba2 or a2 5 2b2
If a2 5 2b2, then the square of a is an even number. Since a2 5 a a and a2 is
even, then a must also be even. An even number can be written as two times
some other integer, so we can write a 5 2k. Substituting into the original
2
equation 2 5 ba2:
(2k) 2
2 5 b2
2
2 5 4k
b2
2b2 5 4k2
b2 5 2k2
This means that b is also even. Therefore, ba is not in simplest form because a and
b have a common factor, 2. This contradicts the assumption that ba is in simplest
form, so !2 must be irrational.
There are many other proofs that !2 is irrational. Research one of these
proofs and discuss your findings with the class.
CUMULATIVE REVIEW
CHAPTERS 1–3
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. Which of the following is not a rational number?
(1) 12
(2) 0.25
(3) !0.04
(4) !0.4
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117
2. When 3a2 2 5a is subtracted from a2 1 4, the difference is
(1) 23a2 1 5a 1 4
(3) 22a2 1 5a 1 4
2
(2) 22a 2 5a 1 4
(4) 2a2 2 5a 2 4
3. When factored completely, 12a2 2 3a is equal to
(1) 3a(4a)
(3) 3(4a2 2 a)
(2) a(12a 2 3)
(4) 3a(4a 2 1)
4. The factors of a2 1 5a 2 6 are
(1) (a 1 3)(a 2 2)
(2) (a 2 3)(a 1 2)
(3) (a 1 6)(a 2 1)
(4) (a 2 6)(a 1 1)
5. If 4(x 1 6) 5 x 2 12, then x is equal to
(1) 212
(2) 26
(3) 6
(4) 12
6. The solution set of y 2 3 5 7 is
(1) {10}
(2) {4, 210}
(3) {24, 10}
(4) {24}
(3) U223, 21 V
(4) U 223, 1 V
(3) 23
(4) 2
7. The solution set of 3x2 2 x 5 2 is
(1) {1, 2}
8. In simplest form,
(1) 83
(2) U23, 1 V
1 1 21
1 2 41
is equal to
(2) 21
9. If x3 2 5 . x8 , then
(1) 21 , x , 1
(2) x . 21 or x , 21
(3) 0 , x , 1
(4) 21 , x , 0
10. In simplest form, !24 1 !150 is equal to
(1) 7 !6
(2) 7 !12
(3) 10 !6
(4) !174
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Divide and write the quotient in simplest form.
a2 2 16
a2 2 a 2 12
12. Solve and check: 2x2 2 9x 5 5.
2
4a
4a 1
2a
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Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Find the solution set of the inequality 3b 1 6 , 7 and graph the solution
on the number line.
14. On a test, the number of questions that Tyler answered correctly was 4
more than twice the number that he answered incorrectly. If the ratio of
correct answers to incorrect answers is 8 : 3, how many questions did Tyler
answer correctly?
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. On her way to work, Rachel travels two miles on local roads and 12 miles
on the highway. Her speed on the highway is twice her speed on local
roads. Her driving time for the trip to work is 16 minutes. What is Rachel’s
rate of speed for each part of the trip?
16. The length of Bill’s garden is 2 yards more than four times the width. The
area of the garden is 30 square yards. What are the dimensions of the
garden?
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CHAPTER
RELATIONS
AND
FUNCTIONS
Long-distance truck drivers keep very careful watch
on the length of time and the number of miles that they
drive each day.They know that this relationship is given
by the formula rt d. On some trips, they are required
to drive for a fixed length of time each day, and
d
r
that makes a constant. When this is the case, we say
that the distance traveled and the rate of travel are
directly proportional. When the rate of travel increases,
the distance increases and when the rate of travel
decreases, the distance decreases. On other trips, truck
drivers are required to drive a fixed distance each day,
and that makes rt equal a constant.When this the case,
we say that the time and the rate of travel are inversely
proportional. When the rate of travel is increased, the
time is decreased and when the rate of travel is
decreased, the time is increased. Each of these cases
defines a function.
In this chapter, we will study some fundamental
properties of functions, and in the chapters that follow,
we will apply these properties to some important
classes of functions.
4
CHAPTER
TABLE OF CONTENTS
4-1 Relations and Functions
4-2 Function Notation
4-3 Linear Functions and Direct
Variation
4-4 Absolute Value Functions
4-5 Polynomial Functions
4-6 The Algebra of Functions
4-7 Composition of Functions
4-8 Inverse Functions
4-9 Circles
4-10 Inverse Variation
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
119
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4-1 RELATIONS AND FUNCTIONS
Every time that we write an algebraic expression such as 2x 1 3, we are defining a set of ordered pairs, (x, 2x 1 3). The expression 2x 1 3 means that for every
input x, there is an output that is 3 more than twice x. In set-builder notation,
we write this set of ordered pairs as {(x, y) : y 5 2x 1 3}. This is read “the set of
ordered pairs (x, y) such that y is equal to 3 more than twice x.” The colon (:)
represents the phrase “such that” and the description to the right of the colon
defines the elements of the set. Another set of ordered pairs related to the algebraic expression 2x 1 3 is {(x, y) : y $ 2x 1 3}.
DEFINITION
A relation is a set of ordered pairs.
The inequality y $ 2x 1 3 defines a relation.
For every real number x, there are many possible
values of y. The set of ordered pairs that make
the inequality true is the union of the pairs that
make the equation y 5 2x 1 3 true and the pairs
that make the inequality y . 2x 1 3 true. This is
an infinite set. We can list typical pairs from the
set but cannot list all pairs.
To show the solution set of the inequality
y $ 2x 1 3 as points on the coordinate plane,
we choose some values of x and y such as (0, 3),
(2, 7), and (4, 11) that make the equality true and
draw the line through these points. All of the
pairs on the line and above the line make the
inequality y $ 2x 1 3 true.
The equation y 5 2x 1 3 determines a set of
ordered pairs. In this relation, for every value of
x, there is exactly one value of y. This relation is
called a function.
y
1
O
1
x
DEFINITION
A function is a relation in which no two ordered pairs have the same first element.
The ordered pairs of the function y 5 2x 1 3 depend on the set from which
the first element, x, can be chosen. The set of values of x is called the domain of
the function. The set of values for y is called the range of the function.
DEFINITION
A function from set A to set B is the set of ordered pairs (x, y) such that every
x [ A is paired with exactly one y [ B.
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121
Relations and Functions
The set A is the domain of the function. However, the range of the function
is not necessarily equal to the set B. The
range is the set of values of y, a subset of
B. The symbol { means “is an element
of,” so x [ A means “x is an element of
set A.”
For example, let {(x, y) : y 5 2x 1 3}
be a function from the set of integers to
the set of integers. Every integer is a
value of x but not every integer is a value
of y. When x is an integer, 2x 1 3 is an
odd integer. Therefore, when the domain
of the function {(x, y) : y 5 2x 1 3} is the
set of integers, the range is the set of odd
integers.
x
1
y
0
1
2
0
3
4
1
5
6
2
A
(Integers)
x
7
B
(Integers)
y
1
1
0.5
4
5
4
23
5
2
22 3
5
25 3
A
(Real numbers)
B
(Real numbers)
On the other hand, let {(x, y) : y 5 2x 1 3} be a function from set A to set
B. Let set A be the set of real numbers and set B be the set of real numbers. In
this case, every real number can be written in the form 2x 1 3 for some real
number x. Therefore, the range is the set of real numbers. We say that the function {(x, y) : y 5 2x 1 3} is onto when the range is equal to set B.
DEFINITION
A function from set A to set B is onto if the range of the function is equal to
set B.
The following functions have been studied in earlier courses. Unless otherwise stated, each function is from the set of real numbers to the set of real numbers. In general, we will consider set A and set B to be the set of real numbers.
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Relations and Functions
Constant Function
Linear Function
y 5 a, with a a constant.
y 5 mx 1 b, with m and b constants.
Example: y 5 3
Example: y 5 2x 1 3
y
y
1
1
O 1
x
O 1
The range is {3}.
The function is not onto.
x
The range is the set of real numbers.
The function is onto.
Quadratic Function
Absolute Value Function
y 5 ax2 1 bx 1 c with a, b, and c
constants and a 0.
y 5 a 1 |bx 1 c| with a, b, and
c constants.
Example: y 5 x2 2 4x 1 3
Example: y 5 22 1 |x 1 1|
y
y
1
O
1
O
1
1 x
x
The range is the set of real
numbers $ 21.
The function is not onto.
The range is the set of real
numbers $ 22.
The function is not onto.
Square Root Function
y 5 !ax 1 b, with ax 1 b $ 0.
• The domain is the set of real numbers greater than or equal to 2b
a .
• The range is the set of non-negative real numbers.
y
Example: y 5 !x
This is a function from the set of non-negative real
numbers to the set of non-negative real numbers.
The range is the set of non-negative real numbers.
The function is onto.
1
O 1
x
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Relations and Functions
123
Vertical Line Test
A relation is a function if no two pairs of the relation have the same first element. On a graph, two points lie on the same vertical line if and only if the coordinates of the points have the same first element. Therefore, if a vertical line
intersects the graph of a relation in
y
two or more points, the relation is not
a function. This is referred to as the
vertical line test.
The graph of the relation
1
{(x, y) : y 5 6 !x and x $ 0} is shown
at the right. The graph shows that the
x
1O
line whose equation is x 5 2 intersects
the graph in two points. Every line
whose equation is x = a for a . 0 also
intersects the graph of the relation in
two points. Therefore, the relation is
not a function.
x=2
14411C04.pgs
EXAMPLE 1
For each given set, determine:
a. Is the set a relation?
b. Is the set a function?
(1) {(x, y) : y . x} if x [ {real numbers}
(2) {x : x . 4} if x [ {real numbers}
(3) {(x, y) : y 5 x} if x [ {real numbers}
Solution (1) a. {(x, y) : y . x} is a relation because it is a set of ordered pairs.
b. {(x, y) : y . x} is not a function because many ordered pairs have the
same first elements. For example, (1, 2) and (1, 3) are both ordered
pairs of this relation.
(2) a. {x : x . 4} is not a relation because it is not a set of ordered pairs.
b. {x : x . 4} is not a function because it is not a relation.
(3) a. {(x, y) : y 5 x} is a relation because it is a set of ordered pairs.
b. {(x, y) : y 5 x} is a function because for every real number x there
is exactly one real number that is the absolute value. Therefore, no
two pairs have the same first element.
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Relations and Functions
In part (3) of Example 1, the graph of y 5 x passes the vertical line test,
which we can see using a graphing calculator.
ENTER:
Y
MATH
1
X,T,,n
GRAPH
DISPLAY:
EXAMPLE 2
The graph of a relation is shown at the right.
y
a. Determine whether or not the graph represents a function.
b. Find the domain of the graph.
c. Find the range of the graph.
Solution a. The relation is not a function. Points (1, 0)
and (1, 5) as well as many other pairs have
the same first element. Answer
1
O
1
x
b. The domain is {x : 0 # x # 5}. Answer
c. The range is {y : 0 # y # 5}. Answer
EXAMPLE 3
An artist makes hand-painted greeting cards that she sells online for three dollars each. In addition, she charges three dollars for postage for each package of
cards that she ships, regardless of size. The total amount, y, that a customer pays
for a package of cards is a function of the number of cards purchased, x.
a. Write the relation described above as a function in set-builder notation.
b. What is the domain of the function?
c. Write four ordered pairs of the function.
d. What is the range of the function?
e. Is the function onto?
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Relations and Functions
125
Solution a. The price per card is $3.00, so x cards cost 3x. The cost of postage, $3.00, is a
constant. Therefore, the total cost, y, for x cards is 3x 1 3. In set-builder
notation, this is written as:
{(x, y) : y 5 3x 1 3}
b. The first element, x, which represents the number of cards shipped, must be
a positive integer. Therefore, the domain is the set of positive integers.
c. Many ordered pairs are possible. For example:
3x 1 3
y
(x, y)
1 3(1) 1 3
6
(1, 6)
2 3(2) 1 3
9
(2, 9)
3 3(3) 1 3 12
(3, 12)
4 3(4) 1 3 15
(4, 15)
x
d. Since 3x 1 3 is a multiple of 3, the range is the set of positive integers
greater than 3 that are multiples of 3.
e. The domain is the set of positive integers. This is a function from the set of
positive integers to the set of positive integers. But the range is the set of
positive multiples of 3. The function is not onto because the range is not
equal to the set of positive integers.
EXAMPLE 4
Find the largest domain of each function:
1
a. y 5 x 1
5
1
b. y 5 x2 2x 1
2x 2 3
c. y 5 5x2 x1 1
Solution The functions are well defined wherever the denominators are not equal to 0.
Therefore, the domain of each function consists of the set real numbers such
that the denominator is not equal to 0.
a. x 1 5 5 0 or x 5 25. Therefore, the domain is {x : x 25}.
x2 2 2x 2 3 5 0
b.
(x 2 3)(x 1 1) 5 0
x2350
x1150
x53
x 5 21
Therefore, the domain is {x : x 21, 3}.
c. The denominator 5x2 1 1 cannot be equal to zero. Therefore, the domain is
the set of real numbers.
Answers a. Domain 5 {x : x 25}
c. Domain 5 {Real numbers}
b. Domain 5 {x : x 21, 3}
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Exercises
Writing About Mathematics
1. Explain why {(x, y) : x 5 y2} is not a function but {(x, y) : !x 5 y} is a function.
2. Can y 5 !x define a function from the set of positive integers to the set of positive integers? Explain why or why not.
Developing Skills
In 3–5: a. Explain why each set of ordered pairs is or is not a function. b. List the elements of the
domain. c. List the elements of the range.
3. {(1, 1), (2, 4), (3, 9), (4, 16)}
4. {(1, 21), (0, 0), (1, 1)}
5. {(22, 5), (21, 5), (0, 5), (1, 5), (2, 5)}
In 6–11: a. Determine whether or not each graph represents a function. b. Find the domain for each
graph. c. Find the range for each graph.
6.
1
9.
7.
y
O
1
8.
y
y
1
O
x
10.
y
1
1
x
O
11.
y
1
x
y
1
O 1
x
1
O
1
1
x
O
1
x
In 12–23, each set is a function from set A to set B. a. What is the largest subset of the real numbers
that can be set A, the domain of the given function? b. If set A 5 set B, is the function onto? Justify
your answer.
12. {(x, y) : y 5 2183}
15. {(x, y) : y 5 2x2 1 3x 2 2}
18. U (x, y) : y 5 x1 V
21. U (x, y) : y 5 x2 11 1 V
13. {(x, y) : y 5 5 2 x }
14. {(x, y) : y 5 x2}
16. E (x, y) : y 5 !2x F
17. {(x, y) : y 5 4 2 x}
1
22. U (x, y) : y 5 xx 1
2 1V
5
23. U (x, y) : y 5 |xx 2
2 3| V
19. E (x, y) : y 5 !3 2 xF
20. e (x, y) : y 5 !x 11 1 f
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Function Notation
127
Applying Skills
24. The perimeter of a rectangle is 12 meters.
a. If x is the length of the rectangle and y is the area, describe, in set-builder notation, the
area as a function of length.
b. Use integral values of x from 0 to 10 and find the corresponding values of y. Sketch
the graph of the function.
c. What is the largest subset of the real numbers that can be the domain of the function?
25. A candy store sells candy by the piece for 10 cents each. The amount that a customer pays
for candy, y, is a function of the number of pieces purchased, x.
a. Describe, in set-builder notation, the cost in cents of each purchase as a function of the
number of pieces of candy purchased.
b. Yesterday, no customer purchased more than 8 pieces of candy. List the ordered pairs
that describe possible purchases yesterday.
c. What is the domain for yesterday’s purchases?
d. What is the range for yesterday’s purchases?
4-2 FUNCTION NOTATION
The rule that defines a function may be expressed in many ways. The letter f, or
some other lowercase letter, is often used to designate a function. The symbol
f(x) represents the value of the function when evaluated for x. For example, if f
is the function that pairs a number with three more than twice that number, f
can also be written in any one of the following ways:
1. f 5 {(x, y) : y 5 2x 1 3}
The function f is the set of ordered pairs
(x, 2x 1 3).
2. f(x) 5 2x 1 3
The function value paired with x is 2x 1 3.
3. y 5 2x 1 3
The second element of the pair (x, y) is 2x 1 3.
4. {(x, 2x 1 3)}
The set of ordered pairs whose second element
is 3 more than twice the first.
5. f: x → 2x 1 3
Under the function f, x maps to 2x 1 3.
f
6. x S 2x 1 3
The image of x for the function f is 2x 1 3.
The expressions in 2, 3, and 4 are abbreviated forms of that in 1. Note that
y and f(x) are both symbols for the second element of an ordered pair of the
function whose first element is x. That second element is often called the function value. The equation f(x) 5 2x 1 3 is the most common form of function
notation. For the function f 5 {(x, y) : y 5 2x 1 3}, when x 5 5, y 5 13 or
f(5) 5 13. The use of parentheses in the symbol f(5) does not indicate multipli-
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Relations and Functions
cation. The symbol f(5) represents the function evaluated at 5, that is, f(5) is the
second element of the ordered pair of the function f when the first element is 5.
In the symbol f(x), x can be replaced by any number in the domain of f or by
any algebraic expression that represents a number in the domain of f.
Therefore, if f(x) 5 2x 1 3:
1. f(24) 5 2(24) 1 3 5 25
2. f(a 1 1) 5 2(a 1 1) 1 3 5 2a 1 5
3. f(2x) 5 2(2x) 1 3 5 4x 1 3
4. f(x2) 5 2x2 1 3
We have used f to designate a function throughout this section. Other
letters, such as g, h, p, or q can also be used. For example, g(x) 5 2x defines a
function. Note that if f(x) 5 2x 1 3 and g(x) 5 2x, then f(x) and g(x) 1 3 define
the same set of ordered pairs and f(x) 5 g(x) 1 3.
EXAMPLE 1
Let f be the set of ordered pairs such that the second element of each pair is 1
more than twice the first.
a. Write f(x) in terms of x.
b. Write four different ways of symbolizing the function.
c. Find f(7).
Solution a. f(x) 5 1 1 2x
b. {(x, y) : y 5 1 1 2x}
f
x S 1 1 2x
y 5 1 1 2x
f: x → 1 1 2x
c. f(7) 5 1 1 2(7) 5 15
Exercises
Writing About Mathematics
1. Let f(x) 5 x2 and g(x 1 2) 5 (x 1 2)2. Are f and g the same function? Explain why or why
not.
2. Let f(x) 5 x2 and g(x 1 2) 5 x2 1 2. Are f and g the same function? Explain why or why
not.
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Function Notation
129
Developing Skills
In 3–8, for each function: a. Write an expression for f(x). b. Find f(5).
3. y = x 2 2
6. {(x, 5x)}
f
5. f: x → |3x 2 7|
4. x S x2
7. y 5 !x 2 1
f
8. x S x2
In 9–14, y 5 f(x). Find f(23) for each function.
9. f(x) 5 7 2 x
12. f(x) 5 4
10. y 5 1 1 x2
11. f(x) 5 24x
13. y 5 !1 2 x
15. The graph of the function f is shown at
the right.
14. f(x) 5 |x 1 2|
y
5
4
3
2
Find:
a. f(2)
b. f(25)
c. f(22)
d.
f A 312 B
5 43 2
O
1 2 3 4 5 x
2
3
4
Applying Skills
16. The sales tax t on a purchase is a function of the amount a of the purchase. The sales tax
rate in the city of Eastchester is 8%.
a. Write a rule in function notation that can be used to determine the sales tax on a purchase in Eastchester.
b. What is a reasonable domain for this function?
c. Find the sales tax when the purchase is $5.00.
d. Find the sales tax when the purchase is $16.50.
17. A muffin shop’s weekly profit is a function of the number of muffins m that it sells. The
equation approximating the weekly profit is f(m) 5 0.60m 2 900.
a. Draw the graph showing the relationship between the number of muffins sold and the
profit.
b. What is the profit if 2,000 muffins are sold?
c. How many muffins must be sold for the shop to make a profit of $900?
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Relations and Functions
4-3 LINEAR FUNCTIONS AND DIRECT VARIATION
y
1
The graph of the linear function
f 5 {(x, y) : y 5 2x 2 1} is shown to the right.
• The graph has a slope of 2, that is, the ratio
of the change in y to the change in x is 2 : 1.
• The domain and range of this function are
both the set of real numbers.
2x
• The graph has a y-intercept of 21, that is, it
intersects the y-axis at (0, 21).
y=
14411C04.pgs
1
O
1
(0, 1)
x
Since it is a function, no two pairs have the
same first element. We know that this is true
because no vertical line can intersect the graph of
f in more than one point. Note also that no horizontal line can intersect the graph of f in more
than one point. This is called the horizontal line
test. All points with the same second element lie
on the same horizontal line, that is, if f(x1) 5 f(x2), then x1 5 x2. Therefore, for
the function f, no two pairs have the same second element. We say that the function f is a one-to-one function. Every element of the domain is paired with
exactly one element of the range and every element of the range is paired with
exactly one element of the domain. One-to-one can also be written as 1-1.
DEFINITION
A function f is said to be one-to-one if no two ordered pairs have the same
second element.
For any linear function from the set of real numbers to the set of real numbers, the domain and range are the set of real numbers. Since the range is equal
to the set of real numbers, a linear function is onto.
Every non-constant linear function is one-to-one and onto.
EXAMPLE 1
Determine if each of the following functions is one-to-one.
a. y 5 4 2 3x
b. y 5 x2
c. y 5 |x 1 3|
Solution A function is one-to-one if and only if no two pairs of the function have the
same second element.
a. y 5 4 2 3x is a linear function whose graph is a slanted line. A slanted line
can be intersected by a horizontal line in exactly one point. Therefore,
y 5 4 2 3x is a one-to-one function.
b. y = x2 is not one-to-one because (1, 1) and (21, 1) are two ordered pairs of
the function.
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Linear Functions and Direct Variation
c. y 5 |x 1 3| is not one-to-one because (1, 4) and (27, 4) are two ordered
pairs of the function.
EXAMPLE 2
y=
Sketch the graph of y 5 22x 1 3 and show that no two points have the same
second coordinate.
y
Solution The graph of y 5 22x 1 3 is a line with a
y-intercept of 3 and a slope of 22. Two
y=5
points have the same second coordinate if
and only if they lie on the same horizontal
(0, 3)
line. A horizontal line intersects a slanted
line in exactly one point. In the diagram,
2
the graph of y 5 5 intersects the graph of
1
y 5 22x 1 3 in exactly one point. The func2
tion passes the horizontal line test. No two
x
O
points on the graph of y 5 22x 1 3 have
1
the same second element and the function
is one-to-one.
x+
2
3
4
)
2x
h(x
g(x
)
3
2x
)
f(x
)
y
2x
y
1
1
1
2(x
f(x 1)
)
h(x
2x
)
2(x
2
)
Transformations of Linear Functions
g(x
14411C04.pgs
x
1
x
Compare the graphs of f(x) 5 2x,
g(x) 5 2x 1 3, and h(x) 5 2x 2 4.
Compare the graphs of f(x) 5 2x,
g(x) 5 2(x 1 1), and h(x) 5 2(x 2 2).
1. g(x) 5 f(x) 1 3. The graph of g is
the graph of f shifted 3 units up.
1. g(x) 5 f(x 1 1). The graph of g is
the graph of f shifted 1 unit to
the left in the x direction.
2. h(x) 5 f(x) 2 4. The graph of g is
the graph of f shifted 4 units
down.
2. h(x) 5 f(x 2 2). The graph of g is
the graph of f shifted 2 units to
the right in the x direction.
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Relations and Functions
y
y
x)
3x
132
2:18 PM
g(x)
14411C04.pgs
x
f(
)
h(x
1
1
1
x
1
)
x
f(
x
1x
3
x
x)
g(
x
Compare the graphs of f(x) 5 x
and g(x) 5 2x.
Compare the graphs of f(x) 5 x,
g(x) 5 3x, and h(x) 5 31x.
• g(x) 5 2f(x). The graph of g is
the graph of f reflected in the
x-axis.
1. g(x) 5 3f(x). The graph of g is
the graph of f stretched vertically
by a factor of 3.
2. h(x) 5 31f(x). The graph of h is
the graph of f compressed vertically by a factor of 13.
Recall that in general, if f(x) is a linear function, then:
The graph of f(x) 1 a is the graph of f(x) moved |a| units up when a is pos-
itive and |a| units down when a is negative.
The graph of f(x 1 a) is the graph of f(x) moved |a| units to the left when a
is positive and |a| units to the right when a is negative.
The graph of 2f(x) is the graph of f(x) reflected in the x-axis.
When a + 1, the graph of af(x) is the graph of f(x) stretched vertically by a
factor of a.
When 0 * a * 1, the graph of af(x) is the graph of f(x) compressed verti-
cally by a factor of a.
Direct Variation
If a car moves at a uniform rate of speed, the ratio of the distance that the car
travels to the time that it travels is a constant. For example, when the car moves
at 45 miles per hour,
d
t
5 45
where d is distance in miles and t is time in hours.
The equation dt 5 45 can also be written as the function {(t, d) : d 5 45t}.
This function is a linear function whose domain and range are the set of real
numbers. The function is one-to-one and onto. Although the function d 5 45t
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133
can have a domain and range that are the set of real numbers, when t represents
the time traveled, the domain must be the set of positive real numbers. If the car
travels no more than 10 hours a day, the domain would be 0 # t # 10 and the
range would be 0 # d # 450.
When the ratio of two variables is a constant, we say that the variables are
directly proportional or that the variables vary directly. Two variables that are
directly proportional increase or decrease by the same factor. Every direct variation of two variables is a linear function that is one-to-one.
EXAMPLE 3
Jacob can type 55 words per minute.
a. Write a function that shows the relationship between the number of words
typed, w, and the number of minutes spent typing, m.
b. Is the function one-to-one?
c. If Jacob types for no more than 2 hours at a time, what are the domain and
range of the function?
words
w
5m
Solution a. “Words per minute” can be written as minutes
5 55. This is a direct
variation function that can be written as {(m, w) : w 5 55m}.
b. The function is a linear function and every linear function is one-to-one.
c. If Jacob types no more than 2 hours, that is, 120 minutes, at a time, the
domain is 0 # m # 120 and the range is 0 # w # 6,600.
Exercises
Writing About Mathematics
1. Megan said that if a . 1 and g(x) 5 a1f(x), then the graph of f(x) is the graph of g(x)
stretched vertically by the factor a. Do you agree with Megan? Explain why or why not.
2. Kyle said that if r is directly proportional to s, then there is some non-zero constant, c, such
that r 5 cs and that {(s, r)} is a one-to-one function. Do you agree with Kyle? Explain why
or why not.
Developing Skills
In 3–6, each set represents a function. a. What is the domain of each function? b. What is the range
of each function? c. Is the function one-to-one?
3. {(1, 4), (2, 7), (3, 10), (4, 13)}
4. {(0, 8), (2, 6), (4, 4), (6, 2)}
5. {(2, 7), (3, 7), (4, 7), (5, 7), (6, 7)}
6. {(0, 3), (21, 5), (22, 7), (23, 9), (24, 11)}
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In 7–12, determine if each graph represents a one-to-one function.
7.
8.
9.
y
O
y
y
x
O
x
x
O
10.
y
y
11.
12.
y
O
O
x
x
x
O
In 13–20: a. Graph each function. b. Is the function a direct variation? c. Is the function one-to-one?
13. y 5 3x
14. y 5 6 2 x
15. y 5 2x
17. y 5 12x
18. y 5 !x
19. x 5 2
y
16. y 5 x8
20. xy 5 2
21. Is every linear function a direct variation?
22. Is the direct variation of two variables always a linear function?
Applying Skills
In 23–28, write an equation of the direct variation described.
23. The cost of tickets, c, is directly proportional to the number of tickets purchased, n. One
ticket costs six dollars.
24. The distance in miles, d, that Mr. Spencer travels is directly proportional to the length of
time in hours, t, that he travels at 35 miles per hour.
25. The length of a line segment in inches, i, is directly proportional to the length of the segment in feet, f.
26. The weight of a package of meat in grams, g, is directly proportional to the weight of the
package in kilograms, k.
27. Water is flowing into a swimming pool at the rate of 25 gallons per minute. The number of
gallons of water in the pool, g, is directly proportional to the number of minutes, m, that the
pool has been filling from when it was empty.
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135
28. At 9:00 A.M., Christina began to add water to a swimming pool at the rate of 25 gallons per
minute. When she began, the pool contained 80 gallons of water. Christina stopped adding
water to the pool at 4:00 P.M. Let g be the number of gallons of water in the pool and t be
the number of minutes that have past since 9:00 A.M.
a. Write an equation for g as a function of t.
b. What is the domain of the function?
c. What is the range of the function?
d. Is the function one-to-one?
e. Is the function an example of direct variation? Explain why or why not.
Hands-On Activity 1
Refer to the graph of f shown to the right.
1. Sketch the graph of g(x) 5 f(2x).
y
2. Sketch the graph of h(x) 5 2f(x).
3. Describe the relationship between the graph of f(x) and the
graph of g(x).
1
O
1
x
4. Describe the relationship between the graph of f(x) and the
graph of h(x).
5. Sketch the graph of p(x) 5 3f(x).
6. Describe the relationship between the graph of f(x) and the
graph of p(x).
7. Describe the relationship between the graph of f(x) and the graph of 2p(x).
8. Sketch the graph of f(x) 5 x 1 1 and repeat steps 1 through 7.
Hands-On Activity 2
Let f be the function whose graph is shown to the right.
y
1. Sketch the graph of g(x) 5 f(x 1 2).
2. Sketch the graph of h(x) 5 f(x 2 4).
3. Describe the relationship between the graph of f and the
graph of p(x) 5 f(x + a) when a is positive.
1
O
1
x
4. Describe the relationship between the graph of f and the
graph of p(x) 5 f(x + a) when a is negative.
5. Sketch the graph of f(x) 1 2.
6. Sketch the graph of f(x) 2 4.
7. Describe the relationship between the graph of f and the graph of f(x) + a when a is positive.
8. Describe the relationship between the graph of f and the graph of f(x) + a when a is negative.
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Relations and Functions
9. Sketch the graph of 2f(x).
10. Describe the relationship between f(x) and af(x) when a is a constant greater than 0.
11. Sketch the graph of 2f(x).
12. Describe the relationship between f(x) and 2f(x).
13. Sketch the graph of g(x) 5 f(2x).
14. Describe the relationship between f(x) and g(x).
4-4 ABSOLUTE VALUE FUNCTIONS
Let f(x) 5 |x| be an absolute value funcy
tion from the set of real numbers to the
set of real numbers. When the domain is
the set of real numbers, the range of f is
x
the set of non-negative real numbers.
)
Therefore, the absolute value function is
f(x
not onto. For every positive number a,
there are two ordered pairs (a, a) and
1
(2a, a) that are pairs of the function. The
function is not one-to-one. We say that
x
O 1
the function is many-to-one because
many (in this case two) different first elements, a and 2a, have the same second
element a.
We use single letters such as f and g to name a function. For special functions three letters are often used to designate the function. For example, on the
graphing calculator, the absolute value function is named abs. The absolute
value function, abs(, is located in the MATH menu under the NUM heading. For
instance, to evaluate 2(1) 2 5:
ENTER:
DISPLAY:
MATH
1
2
(
1
)
5
)
ENTER
abs(2(1)-5)
3
Note that the opening parenthesis is supplied by the calculator. We enter the
closing parenthesis after we have entered the expression enclosed between the
absolute value bars.
EXAMPLE 1
Graph the function y 5 x 1 3 on a graphing calculator.
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137
Solution To display the graph of y 5 x 1 3, first enter the function value into Y1. Then
graph the function in the ZStandard window.
ENTER:
Y
MATH
X,T,,n
DISPLAY:
3
1
3
ENTER:
ZOOM
6
)
Plot1 Plot2 Plot3
\ Y 1= a b s ( X + 3 )
\ Y 2=
\ Y 3=
\ Y 4=
\ Y 5=
\ Y 6=
\ Y 7=
DISPLAY:
Note that the minimum value of y is 0. The x value when y 5 0 is x 1 3 5 0 or
x 5 23.
The graph of an absolute value function can be used to find the roots of an
absolute value equation or inequality.
EXAMPLE 2
Use a graph to find the solution set of the following:
a. 2x 1 1 5 3
b. 2x 1 1 , 3
c. 2x 1 1 . 3.
1
Solution Draw the graph of y 5 2x 1 1.
2x
y
On the same set of axes, draw the graph of
y 5 3.
y
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a. The points of intersection of the graphs
are (22, 3) and (1, 3), so 2x 1 1 5 3 for
x 5 22 or x 5 1.
y3
1
b. Between x 5 22 and x 5 1, the graph
of y 5 2x 1 1 lies below the graph of
y 5 3, so 2x 1 1 , 3 for 22 , x , 1.
O
1
c. For x , 22 and x . 1, the graph of
y 5 2x 1 1 lies above the graph of
y 5 3, so 2x 1 1 . 3 for x , 22 or x . 1.
Answers a. {22, 1}
b. {x : 22 , x , 1}
c. {x : x , 22 or x . 1}
x
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EXAMPLE 3
Find the coordinates of the ordered pair of the function y 5 4x 2 2 for which
y has a minimum value.
Solution Since y 5 4x 2 2 is always non-negative, its minimum y-value is 0.
When 4x 2 2 5 0, 4x 2 2 5 0. Solve this linear equation for x:
4x 2 2 5 0
4x 5 2
x 5 24 5 12
Calculator Enter the function y 5 4x 2 2 into Y1 and then
Solution press GRAPH . Press 2nd CALC to view the
CALCULATE menu and choose 2: zero. Use the
arrows to enter a left bound to the left of the
minimum, a right bound to the right of the
minimum, and a guess near the minimum. The
calculator will display the coordinates of the
minimum, which is where the graph intersects
the y-axis.
*
Zero
X=.5
Y=0
Answer The minimum occurs at A 12, 0 B .
Exercises
Writing About Mathematics
1. If the domain of the function f(x) 5 3 2 x is the set of real numbers less than 3, is the function one-to-one? Explain why or why not.
2. Eric said that if f(x) 5 2 2 x, then y 5 2 2 x when x # 2 and y 5 x 2 2 when x . 2. Do
you agree with Eric? Explain why or why not.
Developing Skills
In 3–6, find the coordinates of the ordered pair with the smallest value of y for each function.
3. y 5 x
4. y 5 2x 1 8
5. f(x) 5 P x2 2 7 P
6. g(x) 5 5 2 x
In 7–10, the domain of each function is the set of real numbers. a. Sketch the graph of each function.
b. What is the range of each function?
7. y 5 x 2 1
8. y 5 3x 1 9
9. f(x) 5 4x 1 1
10. g(x) 5 P 3 2 x2 P 2 3
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139
Applying Skills
11. A(2, 7) is a fixed point in the coordinate plane. Let B(x, 7) be any point on the same horizontal line. If AB 5 h(x), express h(x) in terms of x.
12. Along the New York State Thruway there are distance markers that give the number of
miles from the beginning of the thruway. Brianna enters the thruway at distance marker
150.
a. If m(x) 5 the number of miles that Brianna has traveled on the thruway, write an
equation for m(x) when she is at distance marker x.
b. Write an equation for h(x), the number of hours Brianna required to travel to distance
marker x at 65 miles per hour
13. a. Sketch the graph of y 5 x.
b. Sketch the graph of y 5 x 1 2
c. Sketch the graph of y 5 x 2 2.
d. Describe the graph of y 5 x 1 a in terms of the graph of y 5 x.
14. a. Sketch the graph of y 5 x.
b. Sketch the graph of y 5 x 1 2.
c. Sketch the graph of y 5 x 2 2.
d. Describe the graph of y 5 x + a in terms of the graph of y 5 x.
15. a. Sketch the graph of y 5 x.
b. Sketch the graph of y 5 2x.
c. Describe the graph of y 5 2x in terms of the graph of y 5 x.
16. a. Sketch the graph of y 5 x.
b. Sketch the graph of y 5 2x.
c. Sketch the graph of y 5 12 x .
d. Describe the graph of y 5 ax in terms of the graph of y 5 x.
17. a. Draw the graphs of y 5 x 1 3 and y 5 5.
b. From the graph drawn in a, determine the solution set of x 1 3 5 5.
c. From the graph drawn in a, determine the solution set of x 1 3 . 5.
d. From the graph drawn in a, determine the solution set of x 1 3 , 5.
18. a. Draw the graphs of y 5 2x 2 4 and y 5 22.
b. From the graph drawn in a, determine the solution set of 2x 2 4 5 22.
c. From the graph drawn in a, determine the solution set of 2x 2 4 . 22.
d. From the graph drawn in a, determine the solution set of 2x 2 4 , 22.
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4-5 POLYNOMIAL FUNCTIONS
Recall that a monomial is a constant, a variable, or the product of constants and
variables. For example, 5x is a monomial in x. A polynomial is a monomial or the
sum of monomials. The expression x3 2 5x2 2 4x 1 20 is a polynomial in x.
A function f is a polynomial function when f(x) is a polynomial in x. For
example, f(x) 5 x3 2 5x2 2 4x 1 20 is a polynomial function.
DEFINITION
A polynomial function f of degree n is a function of the form
f(x) 5 anxn 1 an 2 1xn 2 1 1 c 1 a0, an 0.
The simplest polynomial function is the constant function, y 5 a constant.
For example, y 5 5 is a constant function whose graph is a horizontal line.
The linear function is a polynomial function of degree one. When m 0, the
graph of the linear function y 5 mx 1 b is a slanted line. The y-intercept of the
line is b and the slope of the line is m.
Quadratic Functions
The quadratic function is a polynomial function of degree two. When a 0, the
graph of the quadratic function y 5 ax2 1 bx 1 c is a curve called a parabola.
When a is positive, the parabola opens upward and has a minimum value of
y. When a 5 1, b 5 2, and c 5 23, the quadratic function is y 5 x2 1 2x 2 3.
y
1
O
1
x
x
x2 1 2x 2 3
24
(24)2 1 2(24) 2 3
5
23
(23)2 1 2(23) 2 3
0
22
(22)2 1 2(22) 2 3
23
21
(21)2 1 2(21) 2 3
24
0
(0)2 1 2(0) 2 3
23
1
(1)2 1 2(1) 2 3
0
2
(2)2 1 2(2) 2 3
5
y
The turning point, also called the vertex, for this parabola is (21, 24) and
the axis of symmetry of the parabola is the line x 5 21, the vertical line through
the turning point.
When a is negative, the parabola opens downward and has a maximum value of y. When a 5 21, b 5 22, and c 5 3, the quadratic function is
y 5 2x2 2 2x 1 3.
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141
Polynomial Functions
y
x
2x2 2 2x 1 3
y
24
2(24)2 2 2(24) 1 3
25
23
22
1
21
O
1
x
0
1
2
2
0
2
3
2(23) 2 2(23) 1 3
2(22) 2 2(22) 1 3
2
2(21) 2 2(21) 1 3
4
2
3
2
0
2
25
2(0) 2 2(0) 1 3
2(1) 2 2(1) 1 3
2(2) 2 2(2) 1 3
The turning point for this parabola is (21, 4) and the axis of symmetry is the
line x 5 21, the vertical line through the turning point.
Recall from previous courses that for the parabola that is the graph of
f(x) 5 ax2 1 bx + c:
1. The minimum or maximum value of the parabola occurs at the axis of
symmetry.
2. The formula for the axis of symmetry of the parabola is:
x 5 2b
2a
Transformations of Quadratic Functions
If:
f(x) 5 x2
g(x) 5 (x 1 3)2 1 1
Then: g(x) 5 f(x 1 3) 1 1
The function f(x 1 3) 1 1 moves every
point of f(x) 3 units to the left and 1 unit up.
In particular, since the vertex of f is (0, 0),
the vertex of g is (23, 1). Note that for every
point on f, there is a corresponding point on
g that is 3 units to the left and 1 unit up as
shown in the graph on the right.
f(x) =
x2
The quadratic function g(x) 5 x2 1 6x 1 10
can be written as g(x) 5 (x 1 3)2 1 1.
1
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3)2
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(x
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O
1
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Relations and Functions
How does the graph of g(x) 5 2f(x)
and the graph of h(x) 5 12f(x) compare with
the graph of f(x)? The graphs of f(x) 5 x2,
g(x) 5 2x2, and h(x) 5 12x2 are shown to the
right. Compare points on f(x), g(x), and h(x)
that have the same x-coordinate. The graph of
g is the graph of f stretched vertically by a factor of 2. In other words, the y-value of a point
on the graph of g is twice the y value of the
point on f and the y value of a point on h is
one-half the y value of the point on f.
In general, if f(x) is a quadratic function,
then:
x
1 2
2
h(x)
=
x2
y
f(x) =
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g(x) = 2x 2
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1
O
x
1
The vertex of a quadratic function of the form f(x) 5 a(x 2 h) 2 1 k is (h, k).
The function g(x) 5 f(x 2 h) 1 k moves every point of f(x) zhz units to the
left if h is positive and to the right if h is negative and zkz units up if k is positive and down if k is negative.
When f(x) is multiplied by a when a + 1, the graph is stretched in the verti-
cal direction and when f(x) is multiplied by a when 0 * a * 1, f(x) is compressed in the vertical direction.
EXAMPLE 1
Graph y 5 13 (x 2 4) 2 1 3.
Solution The graph of y 5 31 (x 2 4) 2 1 3 is the
graph of y 5 x2 compressed vertically by
a factor of 13, shifted to the right by 4 units
and up by 3 units. The vertex of the
parabola is (4, 3) and it opens upward as
shown in the graph.
y
y 13 (x 4)2 3
1
O
y x2
1
x
Roots of a Polynomial Function
The roots or zeros of a polynomial function are the values of x for which y 5 0.
These are the x-coordinates of the points where the graph of the function
intersects the x-axis. For the function y 5 x2 1 2x 2 3 as well as for the function
y 5 2x2 2 2x 1 3, the roots or zeros of the function are 23 and 1.
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Polynomial Functions
These roots can also be found by letting y 5 0 and solving the resulting quadratic functions for x.
y 5 x2 1 2x 2 3
y 5 2x2 2 2x 1 3
0 5 x2 1 2x 2 3
0 5 2(x2 1 2x 2 3)
0 5 (x 1 3)(x 2 1)
0 5 2(x 1 3)(x 2 1)
x1350
x 5 23
x2150
x1350
x51
x2150
x 5 23
x51
Higher Degree Polynomial Functions
A polynomial function of degree two, a quadratic function, has one turning point. A polynomial function of
degree three has at most two turning points. The graph
at the right is the graph of the polynomial function
y 5 x3 1 x2 2 x 2 1. This graph has a turning point at
x 5 21 and at a point between x 5 0 and x 5 1.
A polynomial function of degree three can have
one or three real roots or zeros. It appears that the polynomial function y 5 x3 1 x2 2 x 2 1 has only two real
roots. However, the root 21, at which the graph is tangent to the x-axis, is called a double root, a root that
occurs twice. We can see that this is true by solving the
polynomial function algebraically. To find the roots of a
polynomial function, let y equal zero.
y
1
O
x
1
y 5 x3 1 x2 2 x 2 1
0 5 x3 1 x2 2 x 2 1
0 5 x2(x 1 1) 2 1(x 1 1)
0 5 (x 1 1)(x2 2 1)
0 5 (x 1 1)(x 1 1)(x 2 1)
x1150
x 5 21
x1150
x 5 21
x2150
x51
The polynomial function of degree three has three real roots, but two of
these roots are equal. Note that if 1 is a root of the polynomial function, (x 2 1)
is a factor of the polynomial, and if 21 is a double root of the polynomial function, (x 1 1)(x 1 1) or (x 1 1)2 is a factor of the polynomial. In general:
If a is a root of a polynomial function, then (x – a) is a factor of the
polynomial.
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Relations and Functions
Note also that the polynomial function y 5 x3 1 x2 2 x 2 1 has three roots.
Any polynomial function of degree three has 3, 2, or 1 real roots. If it has two
real roots, one of them is a double root. These real roots may be irrational numbers. In general:
A polynomial function of degree n has at most n distinct real roots.
Examining the Roots of
Polynomial from the Graph
We found the roots of the polynomial y 5 x3 1 x2 2 x 2 1 by factoring, but not
all polynomials can be factored over the set of integers. However, we can use a
calculator to sketch the graph of the function and examine the roots using the
zero function.
For example, to examine the roots of f(x) 5 x4 2 4x3 2 x2 1 16x 2 12, enter
the function into Y1 and view the graph in the ZStandard window. The graph
appears to cross the x-axis at 22, 1, 2, and 3. We can verify that these are the
roots of the function.
CALC to view the CALCULATE menu and choose 2: zero.
Press 2nd
Use the arrows to enter a left bound to the left of one of the zero values, a right
bound to the right of the zero value, and a guess near the zero value. The calculator will display the coordinates of the point at which the graph intersects the
x-axis. Repeat to find the other roots.
*
*
Zero
X=-2
Y=0
Zero
X=1
*
Zero
X=2
Y=0
Y=0
*
Zero
X=3
Y=0
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Polynomial Functions
145
We can verify that these are the roots of the function by evaluating the function at each value:
f(22) 5 (22) 4 2 4(22) 3 2 (22) 2 1 16(22) 2 12
5 16 2 4(28) 2 4 2 32 2 12
50
f(2) 5 (2) 4 2 4(2) 3 2 (2) 2 1 16(2) 2 12
5 16 2 4(8) 2 4 1 32 2 12
50
f(1) 5 (1) 4 2 4(1) 3 2 (1) 2 1 16(1) 2 12
5 1 2 4 2 1 1 16 2 12
50
f(3) 5 (3) 4 2 4(3) 3 2 (3) 2 1 16(3) 2 12
5 81 2 4(27) 2 9 1 48 2 12
50
Note: The calculator approach gives exact values only when the root is an integer or a rational number that can be expressed as a finite decimal. The calculator returns approximate values for rational roots with infinite decimal values
and for irrational roots.
EXAMPLE 2
From the graph, determine the roots of the polynomial function p(x).
y
1
O1
x
Solution The graph intersects the x-axis at 0, 3, and 5.
Therefore, the roots or zeros of the function are 0, 3, and 5.
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Relations and Functions
EXAMPLE 3
Find the real roots of f(x) 5 x4 2 2x3 2 2x2 2 2x 2 3 graphically.
Solution Enter the function into Y1 and graph the function into the standard viewing window. The
graph appears to cross the x-axis at 3 and 21.
We can verify that these are the roots of the
function. Press 2nd CALC 2 to use the
zero function. Use the arrows to enter a left
bound to the left of one of the zero values, a
right bound to the right of the zero value, and a
guess near the zero value. Examine one zero
value at a time. Repeat to find the other roots.
*
Zero
X=-1
*
Y=0
Zero
X=3
Y=0
Answer The real roots of the function are 21 and 3.
EXAMPLE 4
Approximate the real roots of
f(x) 5 13 (x3 1 x2 2 11x 2 3) to the
nearest tenth.
y
Solution The graph appears to cross the x1
axis at 23.7, 20.3, and 3. We can
x
O
verify that these are valid approximations of the roots of the function
1
by using the graphing calculator.
Enter the function into Y1 and
graph the function into the
standard viewing window. Press
CALC
2nd
2 to use the zero
function. Use the arrows to enter a left bound to the left of one of the zero
values, a right bound to the right of the zero value, and a guess near the zero
value. Examine one zero value at a time. Repeat to find the other roots.
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Polynomial Functions
*
*
Zero
X=-3.732051
*
Zero
X=-.2679492 Y=0
Y=-3.33E-13
147
Zero
X=3
Y=-1E-12
Answer The real roots of the function are approximately equal to 23.73, 20.27, and 3.
Exercises
Writing About Mathematics
1. Tiffany said that the polynomial function f(x) 5 x4 1 x2 1 1 cannot have real roots. Do you
agree with Tiffany? Explain why or why not.
2. The graph of y 5 x2 2 4x 1 4 is tangent to the x-axis at x 5 2 and does not intersect the
x-axis at any other point. How many roots does this function have? Explain your answer.
Developing Skills
In 3–11, each graph shows a polynomial functions from the set of real numbers to the set of real
numbers.
a. Find the real roots or zeros of the function from the graph, if they exist.
b. What is the range of the function?
c. Is the function onto?
d. Is the function one-to-one?
3.
4.
y
1
O
5.
y
y
1
1
1
x
O
1
x
O
1
x
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6.
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Relations and Functions
7.
y
O 1
O
x
10.
y
1
O
1
x
1
x
O
1
x
1
y
O
1
x
11.
y
1
y
1
1
1
9.
8.
y
O
1
x
In 12–17, use a graph to find the solution set of each inequality.
12. x2 1 2x 2 3 , 0
13. x2 1 4x 1 3 # 0
14. x2 2 x 2 2 . 0
15. x2 2 2x 1 1 , 0
16. 2x2 1 6x 2 5 , 0
17. 2x2 2 3x 1 4 $ 0
Applying Skills
18. The perimeter of a rectangle is 24 feet.
a. If x is the measure of one side of the rectangle, represent the measure of an adjacent
side in terms of x.
b. If y is the area of the rectangle, express the area in terms of x.
c. Draw the graph of the function written in b.
d. What are the dimensions of the rectangle with the largest area?
19. The sum of the lengths of the legs of a right triangle is 20 feet.
a. If x is the measure of one of the legs, represent the measure of the other leg in terms
of x.
b. If y is the area of the triangle, express the area in terms of x.
c. Draw the graph of the function written in b.
d. What are the dimensions of the triangle with the largest area?
20. A polynomial function of degree three, p(x), intersects the x-axis at (24, 0), (22, 0), and
(3, 0) and intersects the y-axis at (0, 224). Find p(x).
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149
21. a. Sketch the graph of y 5 x2.
b. Sketch the graph of y 5 x2 1 2
c. Sketch the graph of y 5 x2 2 3.
d. Describe the graph of y 5 x2 1 a in terms of the graph of y 5 x2.
e. What transformation maps y 5 x2 to y 5 x2 1 a?
22. a. Sketch the graph of y 5 x2.
b. Sketch the graph of y 5 (x 1 2)2.
c. Sketch the graph of y 5 (x 2 3)2.
d. Describe the graph of y 5 (x + a)2 in terms of the graph of y 5 x2.
e. What transformation maps y 5 x2 to y 5 (x 1 a)2?
23. a. Sketch the graph of y 5 x2.
b. Sketch the graph of y 5 2x2.
c. Describe the graph of y 5 2x2 in terms of the graph of y 5 x2.
d. What transformation maps y 5 x2 to y 5 2x2?
24. a. Sketch the graph of y 5 x2.
b. Sketch the graph of y 5 3x2
c. Sketch the graph of y 5 31x2.
d. Describe the graph of y 5 ax2 in terms of the graph of y 5 x2 when a . 1.
e. Describe the graph of y 5 ax2 in terms of the graph of y 5 x2 when 0 , a , 1.
25. For the parabola whose equation is y 5 ax2 1 bx 1 c, the equation of the axis of symmetry
is x 5 2b
2a . The turning point of the parabola lies on the axis of symmetry. Therefore its
x-coordinate is 2b
2a . Substitute this value of x in the equation of the parabola to find the
y-coordinates of the turning point. Write the coordinates of the turning point in terms of
a, b, and c.
4-6 THE ALGEBRA OF FUNCTIONS
Function Arithmetic
Fran and Greg work together to produce decorative vases. Fran uses a potter’s
wheel to form the vases and Greg glazes and fires them. Last week, they completed five jobs. The table on the following page shows the number of hours
spent on each job.
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Hours Worked
Job number
Fran
Greg
Total
1
3
7
10
2
2
5
7
3
4
8
12
4
1
2
3
312
5
112
6
712
Let x equal the job number, f(x) 5 the number of hours that Fran worked
on job x, g(x) 5 the number of hours that Greg worked on job x, and t(x) 5 the
total number of hours worked on job x.
Therefore:
f(1) 5 3
f(2) 5 2
f(3) 5 4
f(4) 5 21
g(1) 5 7
g(2) 5 5
g(3) 5 8
g(4) 5 3
t(1) 5 10
t(2) 5 7
t(3) 5 12
t(4) 5 312
f(5) 5 112
g(5) 5 6
t(5) 5 712
For each function, the domain is the set {1, 2, 3, 4, 5} and t(x) 5 f(x) 1 g(x)
or t(x) 5 (f 1 g)(x). We can perform arithmetic operations with functions. In
general, if f(x) and g(x) are functions, then:
(f 1 g)(x) 5 f(x) 1 g(x)
The domain of (f 1 g) is the set of elements common to the domain of f and of
g, that is, the domain of (f 1 g) is the intersection of the domains of f and g.
For example, if f(x) 5 2x2 and g(x) 5 5x, then:
(f 1 g)(x) 5 f(x) 1 g(x) 5 2x2 1 5x
If the domains of f and g are both the set of real numbers, then the domain
of (f 1 g) is also the set of real numbers.
The difference, product, and quotient of two functions, as well as the product of a constant times a function, can be defined in a similar way.
If f(x) and g(x) are functions, then:
1. The difference of f and g is expressed as:
(f 2 g)(x) 5 f(x) 2 g(x)
Example: Let f(x) 5 2x2 and g(x) 5 5x.
Then (f 2 g)(x) 5 f(x) 2 g(x) 5 2x2 2 5x.
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The Algebra of Functions
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2. The product of f and g is expressed as:
(fg)(x) 5 f(x) ? g(x)
Example: Let f(x) 5 2x2 and g(x) 5 5x.
Then (fg)(x) 5 f(x) ? g(x) 5 2x2(5x) 5 10x3.
3. The quotient of f and g is expressed as:
A gf B (x) 5 g(x)
f(x)
2
f(x)
2x
Example: Let f(x) 5 2x2 and g(x) 5 5x. Then A gf B (x) 5 g(x) 5 2x
5x 5 5 .
4. The product of f and a constant, a, is expressed as:
af(x)
2
Example: Let f(x) 5 2x and a 5 3. Then 3f(x) 5 3(2x2) 5 6x2.
Note: If g(x) 5 a, a constant, then (fg)(x) 5 f(x)g(x) 5 f(x) ? (a) 5 af(x). That is,
if one of the functions is a constant, then function multiplication is the product
of a function and a constant.
If the domain of f and of g is the set of real numbers, then the domain of
(f 1 g), (f 2 g), (fg), and af(x) is the set of real numbers and the domain of A gf B
is the set of numbers for which g(x) 0.
In the examples given above, the domain of f and of g is the set of real numbers. The domain of f 1 g, of f 2 g, and of fg is the set of real numbers. The
domain of gf is the set of real numbers for which g(x) 0, that is, the set of
non-zero real numbers.
EXAMPLE 1
Let f(x) 5 3x and g(x) 5 !x.
a. Write an algebraic expression for h(x) if h(x) 5 2f(x) 4 g(x).
b. What is the largest possible domain for f, 2f, g, and h that is a subset of the
set of real numbers?
c. What are the values of f(2), g(2), and h(2)?
6x
5 !x
? !x
5 6xx!x 5 6 !x
!x
b. The domain of f and of 2f is the set of real numbers.
The domain of g is the set of non-negative real numbers.
The domain of h is the set of positive real numbers, that is, the set of all
numbers in the domains of both f and g for which g(x) 0.
Solution a. h(x) 5
2f(x)
g(x)
5
c. f(2) 5 3(2) 5 6
2(3x)
"x
g(2) 5 !2
h(2) 5 6 !2
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Relations and Functions
Transformations and Function Arithmetic
y
Let h(x) 5 3 be a constant function. The second
element of each ordered pair is the constant, 3.
3
The domain of h is the set of real numbers and
the range is {3}. The graph of h(x) 5 3 is a horx
izontal line 3 units above the x-axis.
x
)
x
(
f
Let g(x) 5 x. The domain of g is the set of
x)
g(
real numbers and the range is the set of nonh(x) 3
negative real numbers.
Let f(x) 5 g(x) 1 h(x) 5 x 1 3. The
1
domain of f(x) is the set of real numbers. For
O 1
x
each value of x, f(x) is 3 more than g(x). The
graph of f(x) is the graph of g(x) translated 3 units in the vertical direction.
In general, for any function g:
If h(x) 5 a, a constant, and f(x) 5 g(x) 1 h(x) 5 g(x) 1 a, then the graph
Let h and g be defined as before.
Let f(x) 5 g(x) ? h(x) 5 gh(x) 5 3x.
The domain of f(x) is the set of real numbers. For each value of x, f(x) is 3 times
g(x). The graph of f(x) is the graph of g(x)
stretched by the factor 3 in the vertical
direction.
In general, for any function g:
x
of f(x) is the graph of g(x) translated a units in the vertical direction.
y
3
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f(x)
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)
x
x
g(
h(x) 3
1
O
x
1
If h(x) 5 a, and a is a positive constant, then f(x) 5 g(x) ? h(x) 5 ag(x). The
graph of f(x) is the graph of g(x) stretched by the factor a in the vertical
direction when a + 1 and compressed by a factor of a when 0 * a * 1.
y
x)
g(
x
x
h(x) 1
x)
f(
Let h(x) 5 21 be a constant function.
The second element of each pair is the constant, 21. The domain of h is the set of real
numbers and the range is {21}.
Let g(x) 5 x. The domain of g is the set
of real numbers and the range is the set of
non-negative real numbers.
x
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The Algebra of Functions
Let f(x) 5 g(x) ? h(x) 5 gh(x) 5 21x 5 2x. The domain of f(x) is the set
of real numbers. For each value of x, f(x) is 21 times g(x), that is, the opposite
of g(x). The graph of f(x) is the graph of g(x) reflected in the x-axis.
In general, for any function g:
If h(x) 5 21, then f(x) 5 g(x) ? h(x) 5 2g(x). The graph of f(x) is the graph
of g(x) reflected in the x-axis.
Horizontal translations will be covered in the next section when we discuss
function composition.
EXAMPLE 2
The graphs of f(x), g(x), and h(x) are shown at the
right. The function g(x) is the function f(x) compressed vertically by the factor 12, and the function
h(x) is the function g(x) moved 3 units up.
If f(x) 5 !x, what is the equation of h(x)?
Solution The vertex of the parabola that is the graph of f(x)
is the point (0, 0). When f(x) is compressed by the
factor 21,
y
h(x)
f(x) 1
O
x
g(x) x
1
g(x) 5 12f(x) 5 12 !x
Answer When g(x) is moved 3 units up, h(x) 5 g(x) 1 3 5 12f(x) 1 3 5 12 !x 1 3.
Note: In Example 2, the order that you perform the
transformations is important. If we had translated vertically first, the resulting function would have been
3
1
1
2 (f(x) 1 3) or 2 !x 1 2 .
y
h(x) f(x) 1
O
1 x
2
3
2
x
1
Exercises
Writing About Mathematics
1. Eric said that if f(x) 5 2 2 x and g(x) 5 x 2 2, then (f 1 g)(x) 5 0. Do you agree with
Eric? Explain why or why not.
2. Give an example of a function g for which 2g(x) g(2x). Give an example of a function f
for which 2f(x) 5 f(2x).
x
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Relations and Functions
Developing Skills
3. Let f 5 {(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)}
and g 5 {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25) (6, 36)}.
a. What is the domain of f?
b. What is the domain of g?
c. What is the domain of (g 2 f)?
d. List the ordered pairs of (g 2 f) in set notation.
e. What is the domain of
g
g
f?
f. List the ordered pairs of f in set notation.
In 4–7, the graph of y 5 f(x) is shown. a. Find f(0) for each function. b. Find x when f(x) 5 0. For
each function, there may be no values, one value, or more than one value. c. Sketch the graph of
y 5 f(x) 1 2. d. Sketch the graph of y 5 2f(x). e. Sketch the graph of y 5 2f(x).
4.
5.
y
1
O
1
1
6.
x
O
7.
y
1
O
y
x
x
1
x
y
1
1
1
O
In 8–13, the domain of f(x) 5 4 2 2x and of g(x) 5 x2 is the set of real numbers and the domain of
h(x) 5 x1 is the set of non-zero real numbers.
a. Write each function value in terms of x.
b. Find the domain of each function.
8. (g 1 f)(x)
11.
g
A f B (x)
9. (g 2 h)(x)
12. (g 1 3f)(x)
10. (gh)(x)
13. (2gf)(x)
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155
Applying Skills
14. When Brian does odd jobs, he charges ten dollars an hour plus two dollars for transportation. When he works for Mr. Atkins, he receives a 15% tip based on his wages for the number of hours that he works but not on the cost of transportation.
a. Find c(x), the amount Brian charges for x hours of work.
b. Find t(x), Brian’s tip when he works for x hours.
c. Find e(x), Brian’s earnings when he works for Mr. Atkins for x hours, if e(x) 5 c(x) 1 t(x).
d. Find e(3), Brian’s earnings when he works for Mr. Atkins for 3 hours.
15. Mrs. Cucci makes candy that she sells for $8.50 a pound. When she ships the candy to outof-town customers, she adds a flat shipping charge of $2.00 plus an additional $0.50 per
pound.
a. Find c(x), the cost of x pounds of candy.
b. Find s(x), the cost of shipping x pounds of candy.
c. Find t(x), the total bill for x pounds of candy that has been shipped, if t(x) 5 c(x) 1 s(x).
d. Find t(5), the total bill for five pounds of candy that is shipped.
16. Create your own function f(x) and show that f(x) 1 f(x) 5 2f(x) . Explain why this result is
true in general.
4-7 COMPOSITION OF FUNCTIONS
In order to evaluate a function such as y 5 !x 1 3, it is necessary to perform
two operations: first we must add 3 to x, and then we must take the square root
of the sum. The set of ordered pairs {(x, y) : y 5 !x 1 3} is the composition of
two functions. If we let f(x) 5 x 1 3 and g(x) 5 !x, we can form the function
h(x) 5 !x 1 3. We evaluate the function h as shown below:
f
x
22
S
f
S
0
S
f
f
g
x13
1
S
g
S
3
S
g
g
1
S
4
S
3
4
f
S
15
4
g
S
5
S
8
S
f
g
!x 1 3
!1 5 1
(x, h(x))
(22, 1)
!3
!4 5 2
15
#4
5
(0, !3)
!15
2
!8 5 2 !2
(1, 2)
Q 34,
!15
2 R
(5, 2 !2)
The function h is a composite function. It is the set of ordered pairs that
result from mapping x to f(x) and then mapping f(x) to g(f(x)). We say that
h(x) 5 g(f(x)) or that h(x) 5 g + f(x). The symbol + is read “of.” The function h
is the composition of g following f. The function f is applied first, followed by g.
We evaluate the composition of functions from right to left.
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Relations and Functions
In the introductory example, g is defined only for non-negative real numbers. However, f is a function from the set of real numbers to the set of real numbers. In order for the composition g + f to be well defined, we must restrict the
domain of f so that its range is a subset of the domain of g. Since the largest possible domain of g is the set of non-negative real numbers, the range of f must be
the non-negative real numbers, that is, f(x) $ 0 or x 1 3 $ 0 and x $ 23.
Therefore, when we restrict f to {x : x $ 23}, the composition g + f is well defined.
The domain of h is {x : x $ 23} and the range of h is {y : y $ 0}.
The domain of the composition g + f is the set of all x in the domain of f
where f(x) is in the domain of g.
EXAMPLE 1
Given: f 5 {(0, 1), (1, 3), (2, 5), (3, 7)} and g 5 {(1, 0), (3, 3), (5, 1), (7, 2)}
Find: a. f + g b. g + f
Solution a. To find (f + g)(x), work from
right to left. First find g(x) and
then find f(g(x)).
x
1
3
5
7
g
S
g
S
g
S
g
S
g
S
g(x)
0
3
1
2
f
S
f
S
f
S
f
S
f
S
b. To find (g + f)(x), work from
right to left. First find f(x) and
then find g(f(x)).
(f + g)(x)
1
7
3
5
x
0
1
2
3
f
S
f
S
f
S
f
S
f
S
f(x)
1
3
5
7
g
S
g
S
g
S
g
S
g
S
(g + f)(x)
0
3
1
2
Answers a. f + g 5 {(1, 1), (3, 7), (5, 3), (7, 5)}
b. g + f 5 {(0, 0), (1, 3), (2, 1), (3, 2)}
EXAMPLE 2
Let h(x) 5 (p + q)(x), p(x) 5 x, and q(x) 5 x 2 4.
a. Find h(23).
b. What is the domain of h?
c. What is the range of h?
Solution a. h(23) 5 p(q(23)) 5 p(23 2 4) 5 p(27) 5 27 5 7
b. Since the range of q is the same as the domain of p, all elements of the
domain of q can be used as the domain of h 5 (p + q). The domain of q is
the set of real numbers, so the domain of h is the set of real numbers.
c. All elements of the range of h 5 (p + q) are elements of the range of p. The
range of p is the set of non-negative real numbers.
Answers a. h(23) 5 7
b. The domain of h is the set of real numbers.
c. The range of h is the set of non-negative real numbers.
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EXAMPLE 3
Show that the composition of functions is not commutative, that is, it is not necessarily true that f(g(x)) 5 g(f(x)).
Solution If the composition of functions is commutative, then f(g(x)) 5 g(f(x)) for all
f(x) and all g(x). To show the composition of functions is not commutative, we
need to show one counterexample, that is, one pair of functions f(x) and g(x)
such that f(g(x)) g(f(x)).
Let f(x) 5 2x and g(x) 5 x 1 3.
f(g(x)) 5 f(x 1 3)
g(f(x)) 5 g(2x)
5 2(x 1 3)
5 2x 1 3
5 2x 1 6
Let x 5 5.
Let x 5 5.
Then f(g(5)) 5 2(5) 1 6 5 16.
Then g(f(5)) 5 2(5) 1 3 5 13.
Therefore, f(g(x)) g(f(x)) and the composition of functions is not commutative.
EXAMPLE 4
Let p(x) 5 x 1 5 and q(x) 5 x2.
a. Write an algebraic expression for r(x) 5 (p + q)(x).
b. Write an algebraic expression for f(x) 5 (q + p)(x).
c. Evaluate r(x) for {x : 22 # x # 3} if x [ {Integers}.
d. Evaluate f(x) for {x : 22 # x # 3} if x [ {Integers}.
e. For these two functions, is (p + q)(x) 5 (q + p)(x)?
Solution a. r(x) 5 (p + q)(x) 5 p(x2) 5 x2 1 5
b. f(x) 5 (q + p)(x) 5 q(x 1 5) 5 (x 1 5)2 5 x2 1 10x 1 25
c.
x
r(x)
(x, r(x))
–2
(22)2 1 5 5 9
(22, 9)
–1
0
1
2
3
2
(21) 1 5 5 6
2
(0) 1 5 5 5
2
(1) 1 5 5 6
2
(2) 1 5 5 9
2
(3) 1 5 5 14
(21, 6)
(0, 5)
(1, 6)
(2, 9)
(3, 14)
d.
x
–2
–1
0
1
2
3
f(x)
(22)2 1 10(22) 1 25 5 9
2
(21) 1 10(21) 1 25 5 16
(x, f(x))
(22, 9)
(21, 16)
2
(0, 25)
2
(1, 36)
2
(2, 49)
2
(3, 64)
(0) 1 10(0) 1 25 5 25
(1) 1 10(1) 1 25 5 36
(2) 1 10(2) 1 25 5 49
(3) 1 10(3) 1 25 5 64
e. No. From the tables, since (p + q)(3) 5 14 and (q + p)(3) 5 64,
(p + q)(x) (q + p)(x).
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Relations and Functions
Answers a. r(x) 5 x2 1 5
b. f(x) 5 x2 1 10x 1 25
c. {(22, 9), (21, 6), (0, 5), (1, 6), (2, 9), (3, 14)}
d. {(22, 9), (21, 16), (0, 25), (1, 36), (2, 49), (3, 64)}
e. No
Transformations and Function Composition
Let p(x) 5 x 1 5 and q(x) 5 x 2 5. The domains of p and q are the set of real
numbers and the ranges are also the set of real numbers.
Let g(x) 5 x2. The domain of g is the set of real numbers and the range is
the set of non-negative real numbers.
Let f(x) 5 (g + q)(x) 5 g(x 1 5) 5 (x 1 5)2.
Let h(x) 5 (g + q)(x) 5 g(x 2 5) 5 (x 2 5)2. The domains of f and h are the
set of real numbers and the ranges are the set of non-negative real numbers.
If we sketch the graph of g(x) 5 x2 and the graph of f(x) 5 (x 1 5)2, note
that f(x) is the graph of g(x) moved 5 units to the left. Similarly, the graph of
h(x) 5 (x 2 5)2 is the graph of g(x) moved 5 units to the right.
g(x)
1
O1
= g(x
h(x)
= x2
x
5)
5)
y
= g(
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f(x)
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x
In general, for any function g:
If h(x) 5 g(x 1 a) for a constant a, the graph of h(x) is the graph of g(x)
moved zaz units to the left when a is positive and zaz units to the right when
a is negative.
Compare the functions and their graphs on the following page.
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Composition of Functions
f(x) 5 !x
y
g(x 1 1) 5 !x 1 1
y
1
O 1
h(x) 5 !x 1 1
1
O 1
x
Domain: x $ 0
x
Domain: x $ 21
p(x) 5 !x 1 1
y
y
1
1
O 1
159
O 1
x
Domain: x $ 0
x
Domain: x $ 21
Note that:
• g(x) is the same function as f(x), x → !x is the same as x 1 1 → !x 1 1.
• h(x) is f(x) moved 1 unit up.
• p(x) is f(x) moved 1 unit to the left.
Exercises
Writing About Mathematics
1. Marcie said that if f(x) 5 x2, then f(a 1 1) 5 (a 1 1)2. Do you agree with Marcie? Explain
why or why not.
2. Explain the difference between fg(x) and f(g(x)).
Developing Skills
In 3–10, evaluate each composition for the given values if f(x) 5 3x and g(x) 5 x 2 2.
3. f(g(4))
4. g(f(4))
7. f(f(5))
8. g(g(5))
5. f + g(22)
9. f A g A 23 B
B
6. g + f(22)
10. g A f A 23 B
B
In 11–18: a. Find h(x) when h(x) 5 g(f(x)). b. What is the domain of h(x)? c. What is the range of
h(x)? d. Graph h(x).
11. f(x) 5 2x 1 1, g(x) 5 4x
12. f(x) 5 3x, g(x) 5 x 2 1
13. f(x) 5 x2, g(x) 5 4 1 x
14. f(x) 5 4 1 x, g(x) 5 x2
15. f(x) 5 x2, g(x) 5 !x
16. f(x) 5 2 1 x, g(x) 5 2x
17. f(x) 5 5 2 x, g(x) 5 x
18. f(x) 5 3x 2 1, g(x) 5 13 (x 1 1)
In 19–22, let f(x) 5 x. Find f(g(x)) and g(f(x)) for each given function.
19. g(x) 5 x 1 3
20. g(x) 5 2x
21. g(x) 5 2x 1 3
22. g(x) 5 5 2 x
23. For which of the functions given in 19–22 does f(g(x)) 5 g(f(x))?
24. If p(x) 5 2 and q(x) 5 x 1 2, find p(q(5)) and q(p(5)).
25. If h(x) 5 2(x 1 1) and h(x) 5 f(g(x)), what are possible expressions for f(x) and for g(x)?
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Applying Skills
26. Let c(x) represent the cost of an item, x, plus sales tax and d(x) represent the cost of an item
less a discount of $10.
a. Write c(x) using an 8% sales tax.
b. Write d(x).
c. Find c + d(x) and d + c(x). Does c + d(x) 5 d + c(x)? If not, explain the difference
between the two functions. When does it makes sense to use each function?
d. Which function can be used to find the amount that must be paid for an item with a
$10 discount and 8% tax?
27. The relationship between temperature and the rate at which crickets chirp can be approximated by the function n(x) 5 4x 2 160 where n is the number of chirps per minute and x is
the temperature in degrees Fahrenheit. On a given summer day, the temperature outside
between the hours of 6 A.M. and 12 P.M. can be modeled by the function f(x) 5 0.55x2 1
1.66x 1 50 where x is the number of hours elapsed from 6 P.M.
a. Find the composite function n + f.
b. What is the rate of chirping at 11 A.M.?
4-8 INVERSE FUNCTIONS
Identity Function
A function of the form y 5 mx + b is a linear function. When we let m 5 1 and b 5 0, the function is
y 5 1x 1 0 or y 5 x. This is a function in which the
second element, y, of each ordered pair is equal to
the first element.
The function y 5 x is called the identity func-
y
y
x
1
O1
x
tion, I, such that I(x) 5 x.
The graph of the identity function is shown to
the right.
DEFINITION
The identity function (I) is the function that maps every element of the
domain to the same element of the range.
The domain of the identity can be the set of real numbers or any subset of
the set of real numbers. The domain and range of the identity function are the
same set.
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161
Recall that for the set of real numbers, the identity for addition is 0 because
for each real number a, a 1 0 5 0 1 a = a. We can say that adding the identity
to any real number a leaves a unchanged.
Similarly, for the set of real numbers, the identity for multiplication is 1
because for each real number a, a ? 1 5 1 ? a 5 a. We can say that multiplying
the identity for multiplication by any real number a leaves a unchanged.
Likewise, the identity function is I(x) because for any function f:
(f + I) 5 (I + f) 5 f
The composition of the identity function with any function f leaves f
unchanged. For example, let f(x) 5 2x 1 3 and I(x) be the identity function, that
is, I(x) 5 x.
(I + f)(x)
f
x
S
0
–1
2
5
x
S
f
S
f
S
f
S
f
S
f
(f + I)(x)
I
f(x)
S
2(0) 1 3 5 3
2(21) 1 3 5 1
2(2) 1 3 5 7
2(5) 1 3 5 13
2x 1 3
S
I
S
I
S
I
S
I
S
I
I
f(x)
x
S
3
1
7
13
2x 1 3
0
–1
2
5
x
S
I
S
I
S
I
S
I
S
I
x
0
21
2
5
x
f
S
f(x)
f
S 2(0) 1 3 5 3
f
S 2(21) 1 3 5 1
f
S 2(2) 1 3 5 7
f
S 2(5) 1 3 5 13
f
S
2x 1 3
(I + f)(x) 5 2x 1 3 and (f + I)(x) 5 2x 1 3,
so (I + f)(x) 5 (f + I)(x).
Let f(x) be any function whose domain and range are subsets of real numbers. Let y = x or I(x) 5 x be the identity function. Then:
I
f
x S f(x) S f(x)
and
I
f
x S x S f(x)
Therefore:
(I + f)(x) 5 (f + I)(x) 5 f(x)
Inverse Functions
In the set of real numbers, two numbers are additive inverses if their sum is
the additive identity, 0, and two numbers are multiplicative inverses if their
product is the multiplicative identity, 1. We can define inverse functions in a
similar way.
The function inverse of f is g if for all x in the domains of (f + g) and of
(g + f), the composition (f + g)(x) 5 (g + f)(x) 5 I(x) 5 x. If g is the function
inverse of f, then f is the function inverse of g.
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DEFINITION
The functions f and g are inverse functions if f and g are functions and
(f + g) 5 (g + f) 5 I.
1
For instance, let f(x) 5 2x 1 1 and g(x) 5 x 2
2 .
(f + g)(x)
g
x
S
–2
S
3
5
x
g
g
S
g
S
g
S
f
g(x)
22 2
2
3 2
2
5 2
2
x
1
1
1
f(g(x)) 5 I(x)
S
f
5 232
S
f
51
2 A 232 B 1 1 5 23
(23, 23)
2(1) 1 1 5 3
(3, 3)
2(2) 1 1 5 5
(5, 5)
S
f
52
S
1
2Ax 2
2 B1
f
2 1
2
S
(x, I(x))
15x
(x, x)
(f + g)(x) 5 x
(g + f)(x)
f
x
S
–2
S
3
f
f
S
f
5
S
x
S
f
g
f(x)
S
2(22) 1 1 5 23
S
2(3) 1 1 5 7
g
g
S
g
2(5) 1 1 5 11
S
2x 1 1
S
g
g(f(x)) 5 I(x)
23 2 1
5 22
2
7 2 1
2 53
11 2 1
55
2
(2x 1 1) 2 1
5
2
(x, I(x))
(22. 22)
(3, 3)
(5, 5)
x
(x, x)
(g + f)(x) 5 x
Since (f + g)(x) 5 x and (g + f)(x) 5 x, f and g are inverse functions.
The symbol f 21 is often used for the inverse of the function f. We showed that
1
when f(x) 5 2x 1 1 and g(x) 5 x 2
2 , f and g are inverse functions. We can write:
f(x) 5 g21(x) 5 2x 1 1
y
1
1
g(x) 5 f21 (x) 5 x 2
or
2
–1
The graph of f is the reflection of
the graph of f over the line y = x. For
every point (a, b) on the graph of f,
there is a point (b, a) on the graph
of f–1. For example, (1, 3) is a point on
f and (3, 1) is a point on f–1.
If f 5 {(x, y) : y 5 2x 1 1}, then
f21 5 {(x, y) : x 5 2y 1 1}. Since it is customary to write y in terms of x, solve
x 5 2y 1 1 for y.
y
2x
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y
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(1, 3)
x
1
x 2
y
(3, 1)
1
O1
x
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Inverse Functions
The inverse of a function can be found
by interchanging x and y, and then
solving for y:
163
y 5 2x 1 1
x 5 2y 1 1
x 2 1 5 2y
x 2 1
2
5y
1
In set-builder notation, f 21 5 U (x,y):y 5 x 2
2 V.
Not every function has an inverse. For instance:
Let f 5 {(23, 9), (22, 4), (21, 1), (0, 0), (1, 1), (2, 4), (3, 9)}.
If we interchange the first and second elements of each pair, we obtain the following set:
g 5 {(9, 23), (4, 22), (1, 21), (0, 0), (1, 1), (4, 2), (9, 3)}
Although g is a relation, it is not a function. Therefore, g cannot be the
inverse function of f.
A function has an inverse function if and only if for every first element there
is exactly one second element and for every second element there is exactly
one first element. (Alternatively, a function has an inverse if and only it is
one-to-one.)
Since the existence of the inverse of a function depends on whether or not
the function is one-to-one, we can use the horizontal line test to determine when
a function has an inverse.
Recall that a function that is not one-to-one is said to be many-to-one.
EXAMPLE 1
Let h(x) 5 2 2 x. Find h21(x) and show that h and h21 are inverse functions.
Solution Let h 5 {(x, y) : y 5 2 2 x}. Then h21 5 {(x, y) : x 5 2 2 y}.
Solve x 5 2 2 y for y
in terms of x.
x522y
x 2 2 5 2y
21(x 2 2) 5 21(2y)
2x 1 2 5 y
h21(x) 5 2x 1 2
Show that h(h21(x)) 5 h–1(h (x)) 5 I(x).
h(h21(x)) 5 h(2x 1 2)
h21(h (x)) 5 h21(2 2 x)
5 2 2 (2x 1 2)
5 2(2 2 x) 1 2
521x22
5 22 1 x 1 2
5x✔
5x✔
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Relations and Functions
EXAMPLE 2
If g(x) 5 13x 1 1, find g21(x).
Solution
How to Proceed
(1) For function g, write y in terms of x:
y 5 13x 1 1
(2) To find the inverse of a function,
interchange x and y. Write the function
g21 by interchanging x and y:
x 5 13y 1 1
(3) Solve for y in terms of x:
x 5 13y 1 1
3x 5 y 1 3
3x 2 3 5 y
Answer g21(x) 5 3x 2 3
Inverse of an Absolute Value Function
If y 5 x is the absolute value function, then to form
an inverse, we interchange x and y. But the relation
x 5 y is not a function because for every non-zero
value of x, there are two pairs with the same first element. For example, if y is 3, then x is 3 or 3 and one
pair of the relation is (3, 3). If y is 23, then x is 23 or
3 and another pair of the relation is (3, 23). The relation has two pairs, (3, 3) and (3, 23), with the same
first element and is not a function. This is shown on
the graph at the right.
y
1 x
O
y (3, 3)
1
x
(3, 3)
When the domain of an absolute value function is the set of real numbers,
the function is not one-to-one and has no inverse function.
EXAMPLE 3
The composite function h 5 f + g maps x to 4 2 x.
a. What is the function g?
b. What is the function f?
c. Find h(1) and h(7).
d. Is the function h one-to-one?
e. Does h have an inverse function?
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Inverse Functions
165
Solution a. To evaluate this function, we first find 4 2 x. Therefore, g(x) 5 4 2 x.
b. After finding 4 2 x, we find the absolute value. Therefore, f(x) 5 x.
c. h(1) 5 4 2 x
h(7) 5 4 2 x
5 4 2 1
5 4 2 7
5 3
5 23
53
53
d. The function is not one-to-one because at least two pairs, (1, 3) and (7, 3),
have the same second element.
e. The function h does not have an inverse because it is not one-to-one.
For every ordered pair (x, y) of a function, (y, x) is an ordered pair of the
inverse. Two pairs of h are (1, 3) and (7, 3). Two pairs of an inverse would
be (3, 1) and (3, 7). A function cannot have two pairs with the same first
element. Therefore, h has no inverse function. Any function that is not
one-to-one cannot have an inverse function.
Inverse of a Quadratic Function
The domain of a polynomial function of degree two (or a quadratic function) is
the set of real numbers. The range is a subset of the real numbers. A polynomial
function of degree two is not one-to-one and is not onto.
A function that is not one-to-one does not have an inverse function under
composition. However, it is sometimes possible to select a subset of the domain
for which the function is one-to-one and does have an inverse for that subset of
the original domain. For example, we can restrict the domain of y 5 x2 1 2x 2 3
to those values of x that are greater than or equal to the x coordinate of the
turning point.
For x $ 21, there is exactly one y value
y
for each x value. Some of the pairs of the
x
function y 5 x2 1 2x 2 3 when x $ 21 are
y
(21, 24), (0, 23), (1, 0), and (2, 5).
3
Then (24, 21), (23, 0), (0, 1), and (5, 2)
2 2y
xy
would be pairs of the inverse function,
O
x
x 5 y2 1 2y 2 3, when y $ 21.
1
1
Note that the inverse is the reflection
of the function over the line y = x.
2x3
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Exercises
Writing About Mathematics
1. Taylor said that if (a, b) is a pair of a one-to-one function f, then (b, a) must be a pair of the
inverse function f21. Do you agree with Taylor? Explain why or why not.
2. Christopher said that f(x) 5 x 2 2 and g(x) 5 x 1 2 are inverse functions after he showed
that f(g(2)) 5 2, f(g(5)) 5 5, and f(g(7)) 5 7. Do you agree that f and g are inverse functions? Explain why or why not.
Developing Skills
In 3–10, find each of the function values when f(x) 5 4x.
3. I(3)
4. I(5)
5. I(22)
6. I(f(2))
7. f(I(3))
8. f(f21(26))
9. f21(f(26))
10. fAf21 A !2 B B
In 11–16, determine if the function has an inverse. If so, list the pairs of the inverse function. If not,
explain why there is no inverse function.
11. {(0, 8), (1, 7), (2, 6), (3, 5), (4, 4)}
12. {(1, 4), (2, 7), (1, 10), (4, 13)}
13. {(0, 8), (2, 6), (4, 4), (6, 2)
14. {(2, 7), (3, 7), (4, 7), (5, 7), (6, 7)}
15. {(21, 3), (21, 5), (22, 7), (23, 9), (24, 11)}
16. {(x, y) : y 5 x2 1 2 for 0 # x # 5}
In 17–20: a. Find the inverse of each given function. b. Describe the domain and range of each given
function and its inverse in terms of the largest possible subset of the real numbers.
17. f(x) 5 4x 2 3
20. f(x) 5 !x
5
19. f(x) 5 x 1
3
18. g(x) 5 x 2 5
21. If f 5 {(x, y) : y 5 5x} is a direct variation function, find f21.
22. If g 5 {(x, y) : y 5 7 2 x}, find g21 if it exists. Is it possible for a function to be its own
inverse?
23. Does y 5 x2 have an inverse function if the domain is the set of real numbers? Justify your
answer.
In 24–26, sketch the inverse of the given function.
24.
25.
y
1
O
26.
y
1
1
1
x
O
y
1
x
O
1
x
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Applying Skills
27. On a particular day, the function that converts American dollars, x, to Indian rupees, f(x), is
f(x) 5 0.2532x. Find the inverse function that converts rupees to dollars. Verify that the
functions are inverses.
28. When the function g(x) 5 x2 1 8x 1 18 is restricted to the interval x $ 4, the inverse is
g21 (x) 5 !x 2 2 2 4.
a. Graph g for values of x $ 4. Graph g21 on the same set of axes.
b. What is the domain of g? What is the range of g?
c. What is the domain of g21? What is the range of g21?
d. Describe the relationship between the domain and range of g and its inverse.
4-9 CIRCLES
A circle is the set of points at a fixed distance from a fixed point. The fixed distance is the radius of the circle and the fixed point is the center of the circle.
The length of the line segment through the center of the circle with endpoints on the circle is a diameter of the circle. The definition of a circle can be
used to write an equation of a circle in the coordinate plane.
Center-Radius Form of the Equation of a Circle
Recall the formula for the distance between two
y
A(x2, y2)
points in the coordinate plane. Let B(x1, y1) and
A(x2, y2) be any two points in the coordinate plane.
c
b
C(x2, y1) is the point on the same vertical line as A
a
and on the same horizontal line as B. The length of
B(x1, y1) C(x2, y1)
the line segment joining two points on the same horO
x
izontal line is the absolute value of the difference of
their x-coordinates. Therefore, BC 5 x1 2 x2.
The length of the line segment joining two points on the same vertical
line is the absolute value of the difference of their y-coordinates. Therefore,
AC 5 y1 2 y2.
Triangle ABC is a right triangle whose hypotenuse is AB. Use the
Pythagorean Theorem with a 5 BC, b 5 AC, and c 5 AB to express AB in terms
of the coordinates of its endpoints.
c2 5 a2 1 b2
(AB)2 5 (BC)2 1 (AC)2
(AB)2 5 (x1 2 x2)2 1 (y1 2 y2)2
AB 5 " (x1 2 x2) 2 1 (y1 2 y2) 2
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Relations and Functions
If d is the distance from A to B, then:
d 5 "(x1 2 x2) 2 1 (y1 2 y2) 2
To write the equation for the circle of radius r
whose center is at C(h, k), let P(x, y) be any point
on the circle. Using the distance formula, CP is
equal to
y
P(x, y)
r
C(h, k)
CP 5 "(x 2 h) 2 1 (y 2 k) 2
However, CP is also equal to the radius, r.
O
x
"(x 2 h) 2 1 (y 2 k) 2 5 r
A "(x 2 h) 2 1 (y 2 k) 2 B 2 5 r2
(x 2 h)2 1 (y 2 k)2 5 r2
This is the center-radius form of the equation of a circle.
For any circle with radius r and center at the origin, (0, 0), we can write the
equation of the circle by letting h 5 0 and k 5 0.
(x 2 h)2 1 (y 2 k)2 5 r2
(x 2 0)2 1 (y 2 0)2 5 r2
x2 1 y2 5 r2
The equation of the circle with radius 3 and
center at the origin is
y
(x 2 0)2 1 (y – 0)2 5 (3)2
1 (0, 0)
O 1
or
2
(x2 y2) 9
x
2
x 1y 59
The graph intersects the x-axis at (23, 0)
and (3, 0) and the y-axis at (0, 2 3) and (0, 3).
The equation of the circle with radius 3 and
center at (4, 25) is
(x 2 4)2 1 (y 2 (25))2 5 32
(4, 5)
(x 4)2 (y 5)2 9
or
2
(x 2 4) 1 (y 1 5)2 5 9
Note that when the graph of the circle with radius 3 and center at the origin
is moved 4 units to the right and 5 units down, it is the graph of the equation
(x 2 4)2 1 (y 1 5)2 5 9.
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Circles
A circle is drawn on the coordinate
plane. We can write an equation of the
circle if we know the coordinates of the
center and the coordinates of one point
on the circle. In the diagram, the center
of the circle is at C(22, 3) and one
point on the circle is P(1, 4). To find the
equation of the circle, we can use the
distance formula to find CP, the radius
of the circle.
169
y
P(1, 4)
C(2, 3)
Q(5, 2)
1
O
1
x
CP 5 "(22 2 1) 2 1 (3 2 4) 2
5 " (23) 2 1 (21) 2
5 !9 1 1
5 !10
Therefore, the equation of the circle is
(x 2 (22))2 1 (y 2 3)2 5 A !10B 2
(x 1 2)2 1 (y 2 3)2 5 10
Note that Q(25, 2) is also a point on the circle. If we substitute the coordinates of Q for x and y in the equation, we will have a true statement.
(x 1 2)2 1 (y 2 3)2 5 10
?
(25 1 2)2 1 (2 2 3)2 5 10
?
(23)2 1 (21)2 5 10
?
9 1 1 5 10
10 5 10 ✔
Standard Form of the Equation of a Circle
The standard form of the equation of a circle is:
x2 1 y2 1 Dx 1 Ey 1 F 5 0
We can write the center-radius form in standard form by expanding the
squares of the binomials.
(x 2 h)2 1 (y 2 k)2 5 r2
x2 2 2hx + h2 1 y2 2 2ky 1 k2 5 r2
This equation in standard form is x2 1 y2 2 2hx 2 2ky 1 h2 1 k2 2 r2 5 0.
Note that in the standard form of the equation of a circle, the coefficients of
x2 and y2 are both equal and usually equal to 1. If we compare the two equations,
D 5 22h, E 5 22k, and F 5 h2 1 k2 2 r2.
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EXAMPLE 1
a. Find the center-radius form of the equation x2 1 y2 2 4x 1 6y 2 3 5 0.
b. Determine the center and radius of the circle.
Solution a. D 5 24 5 22h. Therefore, h 5 2 and h2 5 4.
E 5 6 5 22k. Therefore, k 5 23 and k2 5 9.
F 5 23 5 h2 1 k2 2 r2. Therefore, 23 5 4 1 9 – r2 or r2 5 16.
The equation of the circle in center-radius form is (x 2 2)2 1 (y 1 3)2 5 16.
b. This is a circle with center at (2, 23) and radius equal to !16 5 4.
Answers a. (x 2 2)2 1 (y 1 3)2 5 16
b. Center 5 (2, 23), radius 5 4
EXAMPLE 2
The endpoints of a diameter of a circle are P(6, 1) and Q(24, 25). Write the
equation of the circle: a. in center-radius form b. in standard form.
Solution a. The center of a circle is the midpoint of a diameter. Use the midpoint formula to find the coordinates of the center, M.
M(xm, ym) 5
5
5
A 1 2 2, 1 2 2 B
6 1 (24) 1 1 (25)
A 2 , 2 B
A 22, 24
2 B
x 1 x y 1 y
5 (1, 22)
Use the distance formula to find MP, the radius, r, of the circle.
Let (x1, y1) 5 (6, 1), the coordinates of P and (x2, y2) 5 (1, 22), the coordinates of M.
r 5 MP 5 " (6 2 1) 2 1 (1 2 (22)) 2
5 "52 1 32
5 !25 1 9
5 !34
Substitute the coordinates of the center and the radius of the circle in the
center-radius form of the equation.
(x 2 h)2 1 (y 2 k)2 5 r2
(x 2 1)2 1 (y 2 (22))2 5 ( !34) 2
(x 2 1)2 1 (y 1 2)2 5 34
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171
b. Expand the square of the binomials of the center-radius form of the equation.
(x 2 1)2 1 (y 1 2)2 5 34
x2 2 2x 1 1 1 y2 1 4y 1 4 5 34
x2 1 y2 2 2x 1 4y 1 1 1 4 2 34 5 0
x2 1 y2 2 2x 1 4y – 29 5 0
Answers a. (x 2 1)2 1 (y 1 2)2 5 34
b. x2 1 y2 2 2x 1 4y 2 29 5 0
EXAMPLE 3
In standard form, the equation of a circle is x2 1 y2 1 3x 2 4y 2 14 5 0. Write
the equation in center-radius form and find the coordinates of the center and
radius of the circle.
Solution
How to Proceed
(1) Complete the square of x2 1 3x. We “complete the square” by writing
the expression as a trinomial that is a perfect square. Since
(x 1 a)2 5 x2 1 2ax 1 a2, the constant term needed to complete the
square is the square of one-half the coefficient of x:
2
x2 1 3x 1 A 32 B 5 x2 1 3x 1 94
5 A x 1 32 B
2
(2) Complete the square of y2 2 4y. The constant term needed to complete
the square is the square of one-half the coefficient of y:
2
2
y2 2 4y 1 A 24
2 B 5 y 2 4y 1 4
5 (y 2 2)2
(3) Add 14 to both sides of the
given equation:
(4) To both sides of the equation,
add the constants needed to
complete the squares:
x2 1 y2 1 3x 2 4y 2 14 5 0
x2 1 y2 1 3x 2 4y 5 14
x2 1 3x 1 94 1 y2 2 4y 1 4 5 14 1 94 1 4
(5) Write the equation in centerradius form. Determine the
center and radius of the circle
from the center-radius form:
A x 1 32 B
A x 1 32 B
A x 1 32 B
2
9
16
1 (y 2 2)2 5 56
4 1 4 1 4
2
1 (y 2 2)2 5 81
4
2
2
1 (y 2 2)2 5 A 92 B
Answer The equation of the circle in center-radius form is A x 1 32 B 1 (y 2 2)2 5 A 92 B .
2
The center is at A 232, 2 B and the radius is 92.
2
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Exercises
Writing About Mathematics
1. Is the set of points on a circle a function? Explain why or why not.
2. Explain why (x 2 h)2 1 (y 2 k)2 5 24 is not the equation of a circle.
Developing Skills
In 3–10, the coordinates of point P on the circle with center at C are given. Write an equation of
each circle: a. in center-radius form b. in standard form.
3. P(0, 2), C(0, 0)
4. P(0, 23), C(0, 0)
5. P(4, 0), C(0, 0)
6. P(3, 2), C(4, 2)
7. P(21, 5), C(21, 1)
8. P(0, 23), C(6, 5)
9. P(1, 1), C(6, 13)
10. P(4, 2), C(0, 1)
In 11–19, write the equation of each circle.
11.
12.
y
13.
y
y
1
1
14.
1
O
1
x
15.
y
O
1
O
x
16.
y
1
O
1
y
O
1
x
x
1
x
1
O
17.
x
1
18.
y
19.
y
y
(4, 3)
(2, 1)
1
1
O 1
1
O
1
x
1
(1, 1)
x
(1, 1)
x
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173
In 20–27: a. Write each equation in center-radius form. b. Find the coordinates of the center. c. Find
the radius of the circle.
20. x2 1 y2 2 25 5 0
21. x2 1 y2 2 2x 2 2y 2 7 5 0
22. x2 1 y2 1 2x 2 4y 1 1 5 0
23. x2 1 y2 2 6x 1 2y 2 6 5 0
24. x2 1 y2 1 6x 2 6y 1 6 5 0
25. x2 1 y2 2 8y 5 0
26. x2 1 y2 1 10x 2 5y 2 32 5 0
27. x2 1 y2 1 x 2 3y 2 2 5 0
Applying Skills
28. An architect is planning the entryway into a courtyard as an arch in the shape of a semicircle with a radius of 8 feet. The equation of the arch can be written as (x 2 8)2 1 y2 5 64
when the domain is the set of non-negative real numbers less than or equal to 16 and the
range is the set of positive real numbers less than or equal to 8.
a. Draw the arch on graph paper.
b. Can a box in the shape of a cube whose edges measure 6 feet be moved through the arch?
c. Can a box that is a rectangular solid that measures 8 feet by 8 feet by 6 feet be moved
through the arch?
29. What is the measure of a side of a square
30. What are dimensions of a rectangle whose
that can be drawn with its vertices on
length is twice the width if the vertices
a circle of radius 10?
are on a circle of radius 10?
31. Airplane passengers have been surprised to look down over farmland and see designs, called
crop circles, cut into cornfields. Suppose a farmer wishes to make a design consisting of three
concentric circles, that is, circles with the same center but different radii. Write the equations
of three concentric circles centered at a point in a cornfield with coordinates (2, 2).
Hands-On-Activity
Three points determine a circle. If we know the coordinates of three points, we should be able to determine the equation of the circle through these three points. The three points are the vertices of a triangle. Recall that the perpendicular bisectors of the sides of a triangle meet in a point that is equidistant
from the vertices of the triangle. This point is the center of the circle through the three points.
To find the circle through the points P(5, 3), Q(23, 3), and R(3, 23), follow these steps. Think of
P, Q, and R as the vertices of a triangle.
1. Find the midpoint of PQ.
2. Find the slope of PQ and the slope of the line perpendicular to PQ.
3. Find the equation of the perpendicular bisector of PQ.
4. Repeat steps 1 through 3 for QR.
5. Find the point of intersection of the perpendicular bisectors of PQ and QR. Call this point
C, the center of the circle.
6. Find CP = CQ = CR. The distance from C to each of these points is the radius of the circle.
7. Use the coordinates of C and the radius of the circle to write the equation of the circle.
Show that P, Q, and R are points on the circle by showing that their coordinates make the
equation of the circle true.
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4-10 INVERSE VARIATION
Forty people live in an apartment complex. They often want to divide themselves into groups to gather information that will be of interest to all of the tenants. In the past they have worked in 1 group of 40, 2 groups of 20, 4 groups of
10, 5 groups of 8, 8 groups of 5, and 10 groups of 4. Notice that as the number of
groups increases, the number of people in each group decreases and when the
number of groups decreases, the number of people in each group increases. We
say that the number of people in each group and the number of groups are
inversely proportional or that the two quantities vary inversely.
DEFINITION
Two numbers, x and y, vary inversely or are inversely proportional
when xy 5 c, a non-zero constant.
Let x1 and y1 be two numbers such that x1y1 5 c, a constant. Let x2 and y2 be
two other numbers such that x2y2 5 c, the same constant. Then:
x1
x2
y
5 y2
1
Note that when this inverse variation relationship is written as a proportion,
the order of the terms in x is opposite to the order of the terms in y.
If x and y vary inversely and xy 5 20, then the domain and range of this relation are the set of non-zero real numbers. The graph of this relation is shown
below.
x1y1 5 x2y2
x
or
y
y
20
1
10
2
5
4
4
5
2
10
1
20
1
O 1
x
y
220
21
210
22
25
24
24
25
22
210
21
220
x
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175
Note:
1. No vertical line intersects the graph in more than one point, so the relation is a function.
2. No horizontal line intersects the graph in more than one point, so the
function is one-to-one.
3. This is a function from the set of non-zero real numbers to the set of
non-zero real numbers. The range is the set of non-zero real numbers. The
function is onto.
4. If we interchange x and y, the function remains unchanged. The function is
its own inverse.
5. The graph is called a hyperbola.
EXAMPLE 1
The areas of ABC and DEF are equal. The length of the base of ABC is
four times the length of the base of DEF. How does the length of the altitude
to the base of ABC compare with the length of the altitude to the base of
DEF?
Solution Let b1 be the length of the base and h1 be the height of ABC.
Let b2 be the length of the base and h2 be the height of DEF.
(1) Area of ABC 5 area of DEF.
The lengths of the bases and the
heights of these triangles are
inversely proportional:
(2) The length of the base of ABC
is four times the length of the base
of DEF:
(3) Substitute 4b2 for b1 in the equation
b1h1 5 b2h2:
(4) Divide both sides of the equation
by 4b2:
1
2b1h1
5 12b2h2
b1h1 5 b2h2
b1 5 4b2
4b2 h1 5 b2h2
h1 5 14h2
Answer The height of ABC is one-fourth the height of DEF.
Note: When two quantities are inversely proportional, as one quantity changes
by a factor of a, the other quantity changes by a factor of a1.
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EXAMPLE 2
Mrs. Vroman spends $3.00 to buy cookies for her family at local bakery. When
the price of a cookie increased by 10 cents, the number of cookies she bought
decreased by 1. What was the original price of a cookie and how many did she
buy? What was the increased price of a cookie and how many did she buy?
Solution For a fixed total cost, the number of items purchased varies inversely as the
cost of one item.
Let x1 5 p 5 the original price of one cookie in cents
y1 5 n 5 the number of cookies purchased at the original price
x2 = p 1 10 5 the increased price of one cookie in cents
y2 5 n 2 1 5 the number of cookies purchased at the increased price
(1) Use the product form for
inverse variation:
x1y1 5 x2y2
(2) Substitute values in terms
of x and y:
pn 5 (p 1 10)(n 2 1)
(3) Multiply the binomials:
pn 5 pn 2 p 1 10n 2 10
(4) Solve for p in terms of n:
0 5 2p 1 10n 2 10
p 5 10n 2 10
(5) Since she spends $3 or
300 cents for cookies,
pn 5 300. Write this
equation in terms of n by
substituting 10n 2 10 for p:
(6) Solve for n:
pn 5 300
(10n 2 10)(n) 5 300
10n2 2 10n 5 300
10n2 2 10n 2 300 5 0
n2 2 n 2 30 5 0
(n 2 6)(n 1 5) 5 0
n2650
n1550
n56
(7) Solve for p:
n 5 25 (reject this root)
p 5 10n 2 10
p 5 10(6) 2 10
p 5 50
Answer Originally a cookie cost 50 cents and Mrs. Vroman bought 6 cookies. The price
increased to 60 cents and she bought 5 cookies.
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177
Exercises
Writing About Mathematics
1. Is the function f 5 {(x, y) : xy 5 20} a polynomial function? Explain why or why not.
2. Explain the difference between direct variation and inverse variation.
Developing Skills
In 3–5, each graph is a set of points whose x-coordinates and y-coordinates vary inversely. Write an
equation for each function.
3.
4.
y
5.
y
y
1
1
O1
x
O
1
x
1
O
1
x
In 6–12, tell whether the variables vary directly, inversely, or neither.
6. On his way to work, Randy travels 10 miles at r miles per hour for h hours.
7. A driver travels for 4 hours between stops covering d miles at a rate of r miles per hour.
8. Each day, Jaymee works for 6 hours typing p pages of a report at a rate of m minutes per
page.
9. Each day, Sophia works for h hours typing p pages of a report at a rate of 15 minutes per
page.
10. Each day, Brandon works for h hours typing 40 pages of a report at a rate of m minutes per
page.
11. Each day, I am awake a hours and I sleep s hours.
12. A bank pays 4% interest on all savings accounts. A depositor receives I dollars in interest
when the balance in the savings account is P dollars.
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Applying Skills
13. Rectangles ABCD and EFGH have the same area. The length of ABCD is equal to twice
the length of the EFGH. How does the width of ABCD compare to the width of EFGH?
14. Aaron can ride his bicycle to school at an average rate that is three times that of the rate at
which he can walk to school. How does the time that it takes Aaron to ride to school compare with the time that it takes him to walk to school?
15. When on vacation, the Ross family always travels the same number of miles each day.
a. Does the time that they travel each day vary inversely as the rate at which they travel?
b. On the first day the Ross family travels for 3 hours at an average rate of 60 miles per
hour and on the second day they travel for 4 hours. What was their average rate of
speed on the second day?
16. Megan traveled 165 miles to visit friends. On the return trip she was delayed by construction and had to reduce her average speed by 22 miles per hour. The return trip took 2 hours
longer. What was the time and average speed for each part of the trip?
17. Ian often buys in large quantities. A few months ago he bought several cans of frozen
orange juice for $24. The next time Ian purchased frozen orange juice, the price had
increased by $0.10 per can and he bought 1 less can for the same total price. What was the
price per can and the numbers of cans purchased each time?
CHAPTER SUMMARY
A relation is a set of ordered pairs.
A function is a relation such that no two ordered pairs have the same first
element. A function from set A to set B is the set of ordered pairs (x, y) such
that every x [ A is paired with exactly one y [ B. Set A is the domain of the
function. The range of the function is a subset of set B whose elements are the
second elements of the pairs of the function. A function is onto if the range is
set B.
Function Notation
1. f 5 {(x, y) : y 5 2x 1 3} The function f is the set of ordered pairs
(x, 2x 1 3).
2. f(x) 5 2x 1 3
The function value paired with x is 2x 1 3.
3. y 5 2x 1 3
The second element of the pair (x, y) is 2x 1 3.
4. {(x, 2x 1 3)}
The set of ordered pairs whose second element
is 3 more than twice the first.
5. f: x → 2x 1 3
Under the function f, x maps to 2x 1 3.
f
6. x S 2x 1 3
The image of x for the function f is 2x 1 3.
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Chapter Summary
179
In general, if f(x) and g(x) are functions,
(f 1 g)(x) 5 f(x) 1 g(x)
(f 2 g)(x) 5 f(x) 2 g(x)
(fg)(x) 5 f(x)g(x)
A gf B (x) 5 g(x)
f(x)
The domain of (f 1 g), (f 2 g), and (fg) is the set of elements common to the
domain of f and of g, and the domain of gf is the set of elements common to the
domains of f and of g for which g(x) 0.
A composite function h(x) 5 g(f(x)) is the set of ordered pairs that are
the result of mapping x to f(x) and then mapping f(x) to g(f(x)). The function
h(x) 5 g(f(x)) can also be written as g + f(x). The function h is the composition
of g following f.
The identity function, I(x), is that function that maps every element of the
domain to the same element of the range.
A function f is said to be one-to-one if no two ordered pairs have the same
second element.
Every one-to-one function f(x) has an inverse function f21(x) if
21
f(f (x)) 5 f21(f(x)) 5 I(x). The range of f is the domain of f21 and the range of
f21 is the domain of f.
When the ratio of two variables is a constant, we say that the variables are
directly proportional or that the variables vary directly.
When the domain of an absolute value function, y 5 x, is the set of real
numbers, the function is not one-to-one and has no inverse function.
A polynomial function of degree n is a function of the form
f(x) 5 anx n 1 an21x n21 1 c 1 a0, an 0
The roots or zeros of a polynomial function are the values of x for which
y 5 0. These are the x-coordinates of the points at which the graph of the function intersects the x-axis.
The center-radius form of the equation of a circle is (x 2 h)2 1 (y 2 k)2 5 r2
where (h, k) is the center of the circle and r is the radius. The standard form of
the equation of a circle is:
x2 1 y2 1 Dx 1 Ey 1 F 5 0
Two numbers x and y vary inversely or are inversely proportional when
xy 5 a and a is a non-zero constant. The domain and range of xy = a or y 5 xa
are the set of non-zero real numbers.
For every real number a, the graph of f(x 1 a) is the graph of f(x) moved a
units to the left when a is positive and a units to the right when a is negative.
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For every real number a, the graph of f(x) 1 a is the graph of f(x) moved a
units up when a is positive and a units down when a is negative.
The graph of 2f(x) is the graph of f(x) reflected in the x-axis.
For every positive real number a, the graph of af(x) is the graph of f(x)
stretched by a factor of a in the y direction if a . 1 or compressed by a factor of
a if 0 , a , 1.
VOCABULARY
4-1 Set-builder notation • Relation • Function • Domain of a function •
Range of a function • Function from set A to set B • [ • Onto • Vertical
line test
4-3 Linear Function • Horizontal line test • One-to-one • Directly proportional • Vary directly • Direct variation
4-4 Absolute value function • Many-to-one
4-5 Polynomial function f of degree n • Quadratic function • Parabola •
Turning point • Vertex • Axis of symmetry of the parabola • Roots •
Zeros • Double root
4-7 Composition • Composite function
4-8 Identity function (I) • Inverse functions • f–1
4-9 Circle • Radius • Center • Diameter • Center-radius form of the equation of a circle • Standard form of the equation of a circle • Concentric
circles
4-10 Vary inversely • Inversely proportional • Inverse variation • Hyperbola
REVIEW EXERCISES
In 1–5: Determine if the relation is or is not a function. Justify your answer.
1. {(0, 0), (1, 1), (1, 21), (4, 2), (4, 22), (3, 4), (5, 7)}
2. {(x, y) : y 5 !x }
4. {(x, y) : y 5 2x 1 3}
3. {(x, y) : x2 1 y2 5 9}
5. {(x, y) : y 5 2x4 }
In 6–11, each equation defines a function from the set of real numbers to the set
of real numbers. a. Is the function one-to-one? b. Is the function onto?
c. Does the function have an inverse function? If so, write an equation of the
inverse.
6. y 5 4x 1 3
7. y 5 x2 2 2x 1 5
8. y 5 |x 1 2|
9. y = x3
10. y 5 21
11. y 5 !x 2 4
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181
In 12–17, from each graph, determine any rational roots of the function.
12.
13.
y
y
1
1
O
14.
1
x
O 1
15.
y
y
1
1
O
16.
x
1
17.
y
x
O
1
x
y
1
O
1
1
x
x
O 1
18. What are zeros of the function f(x) 5 x 1 3 2 4?
In 19–22, p 5 {(0, 6), (1, 5), (2, 4), (3, 3), (4, 2)} and q 5 {(0, 1), (1, 3), (2, 5),
(3, 7), (4, 9)}. Find:
19. p 1 q
20. p 2 q
21. pq
p
22. q
In 23–30, for each function f and g, find f(g(x)) and g(f(x)).
23. f(x) 5 2x and g(x) 5 x 1 3
25. f(x) 5 x2 and g(x) 5 x 1 2
27. f(x) 5 x2 and g(x) 5 2 1 3x
7
29. f(x) 5 2x 1 7 and g(x) 5 x 2
2
24. f(x) 5 x and g(x) 5 4x 2 1
26. f(x) 5 !x and g(x) 5 5x 2 3
28. f(x) 5 x5 and g(x) 5 10x
3
30. f(x) 5 x3 and g(x) 5 !x
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In 31–36, write the equation of each circle.
31.
32.
y
y
1
1
O
x
1
O
33.
34.
y
y
1
1
36.
y
x
O 1
x
O 1
35.
x
1
y
(0, 1)
(1, 1)
x
(2, 2)
1
O1
37. The graph of the circle
(x 2 2)2 1 (y 1 1)2 5 10 and the line
y 5 x 2 1 are shown in the diagram. Find
the common solutions of the equations and
check your answer.
x
y
1
O
1
x
38. The graph of y 5 x2 is shifted 4 units to the right and 2 units down. Write
the equation of the new function.
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Review Exercises
183
39. The graph of y 5 |x| is stretched in the vertical direction by a factor of 3.
Write an equation of the new function.
40. The graph of y 5 2x 1 3 is reflected in the x-axis. Write an equation of the
new function.
41. The graph of y 5 |x| is reflected in the x-axis and then moved 1 unit to the
left and 3 units up. Write an equation of the new function.
In 42–45:
a. Write an equation of the relation formed by interchanging the elements of
the ordered pairs of each function.
b. Determine if the relation written in a is an inverse function of the given
function. Explain why or why not.
c. If the given function does not have an inverse function, determine, if possible, a domain of the given function for which an inverse function exists.
42. f(x) 5 2x 1 8
44. f(x) 5 x2
43. f(x) 5 x3
45. f(x) 5 !x
Exploration
For this activity, use Styrofoam cones.
The shapes obtained by cutting a cone are called conic sections. The equation of each of these curves is a relation.
1. What is the shape of the edge of the base of a cone? Cut the cone parallel
to the base. What is the shape of the edges of the two cut surfaces?
2. Cut the cone at an angle so that the cut does not intersect the base of the
cone but is not parallel to the base of the cone. What is the shape of the
edges of the cut surfaces?
3. Cut the cone parallel to one slant edge of the cone. What is the shape of
the curved portion of the edges of the cut surfaces?
4. Cut the cone perpendicular to the base of the cone but not through a
diameter of the base. Repeat this cut on a second cone and place the two
cones tip to tip. What is the shape of the edges of the cut surfaces?
5. Cut the cone perpendicular to the base of the cone through a diameter of
the base. Repeat this cut on a second cone and place the two cones tip to
tip. What is the shape of the edges of the cut surfaces?
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CUMULATIVE REVIEW
CHAPTERS 1–4
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. Which of the following is not a real number?
(1) 21
(3) !23
(2) !3
(4) 0.101001000100001. . .
2. In simplest form, 2x(4x2 2 5) 2 (7x3 1 3x) is equal to
(1) x3 2 7x
(3) x3 1 3x 2 5
(2) x3 2 13x
(4) x3 2 3x 2 10
3. The factors of 3x2 2 7x 1 2 are
(1) (3x 2 2)(x 2 1)
(2) (3x 2 1)(x 2 2)
3
4. The fraction !3
is equal to
(1) !3
3
(2) !3
(3) (3x 1 2)(x 1 1)
(4) (3x 1 1)(x 1 2)
(3) 3 !3
(4) 3
5. The solution set of x 1 2 , 5 is
(1) {x : x , 3}
(2) {x : x , 27}
(3) {x : 27 , x , 3}
(4) {x : x , 2 7 or x . 3}
6. The solution set of x2 2 5x 5 6 is
(1) {3, 22}
(2) {23, 2}
(3) {6, 1}
7. In simplest form, the sum !75 1 !27 is
(1) !102
(2) 34 !3
(3) 8 !6
(4) {6, 21}
(4) 8 !3
8. Which of the following is not a one-to-one function?
(1) y = x 1 2
(3) y 5 x1
(2) y 5 x 1 2
(4) y 5 x2
9. For which of the following are x and y inversely proportional?
(1) y 5 4x
(2) y 5 x4
(3) y 5 x2
(4) y 5 x4
3
10. The fraction aa2 2
2 4 is undefined for
(1) a 5 0
(2) a 5 3
(3) a 5 4
(4) a 5 2 and a 5 22
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Cumulative Review
185
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Factor completely: 2x3 1 2x2 2 2x 2 2.
!5
12. Write 33 1
as an equivalent fraction with a rational denominator.
2 !5
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Find the solution set of the inequality x2 2 2x 2 15 , 0.
14. Write in standard form the equation of the circle with center at (2, 23)
and radius 4.
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. a. Sketch the graph of the polynomial function y 5 4x2 2 8x 2 5.
b. Write an equation of the axis of symmetry.
c. What are the coordinates of the turning point?
d. What are the zeros of the function?
2
21a
16. Simplify the rational expression
and list all values of a for which
1 2 a12
the fraction is undefined.
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CHAPTER
5
CHAPTER
TABLE OF CONTENTS
5-1 Real Roots of a Quadratic
Equation
5-2 The Quadratic Formula
5-3 The Discriminant
5-4 The Complex Numbers
5-5 Operations with Complex
Numbers
5-6 Complex Roots of a
Quadratic Equation
5-7 Sum and Product of the
Roots of a Quadratic
Equation
5-8 Solving Higher Degree
Polynomial Equations
5-9 Solutions of Systems of
Equations and Inequalities
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
186
QUADRATIC
FUNCTIONS AND
COMPLEX NUMBERS
Records of early Babylonian and Chinese mathematics provide methods for the solution of a quadratic
equation.These methods are usually presented in terms
of directions for the solution of a particular problem,
often one derived from the relationship between the
perimeter and the area of a rectangle. For example, a list
of steps needed to find the length and width of a rectangle would be given in terms of the perimeter and the
area of a rectangle.
The Arab mathematician Mohammed ibn Musa
al-Khwarizmi gave similar steps for the solution of the
relationship “a square and ten roots of the same amount
to thirty-nine dirhems.” In algebraic terms, x is the root,
x2 is the square, and the equation can be written as
x2 1 10x 5 39. His solution used these steps:
1. Take half of the number of roots : 12 of 10 5 5
2. Multiply this number by itself: 5 3 5 5 25
3. Add this to 39: 25 1 39 5 64
4. Now take the root of this number: !645 8
5. Subtract from this number one-half the number
of roots: 8 2 12 of 10 5 8 2 5 5 3
6. This is the number which we sought.
In this chapter, we will derive a formula for the
solution of any quadratic equation. The derivation of
this formula uses steps very similar to those used by
al-Khwarizmi.
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187
5-1 REAL ROOTS OF A QUADRATIC EQUATION
The real roots of a polynomial function are the
x-coordinates of the points at which the graph of the
function intersects the x-axis. The graph of the polynomial function y 5 x2 2 2 is shown at the right. The
graph intersects the x-axis between 22 and 21 and
between 1 and 2. The roots of 0 5 x2 2 2 appear to be
about 21.4 and 1.4. Can we find the exact values of
these roots? Although we cannot factor x2 2 2 over
the set of integers, we can find the value of x2 and
take the square roots of each side of the equation.
y
y x2 2
1
O 1
x
0 5 x2 2 2
2 5 x2
6 !2 5 x
The roots of the function y 5 x2 2 2 are the irrational numbers 2!2 and !2.
The graph of the polynomial function
y
y 5 x2 1 2x 2 1 is shown at the right. The graph
shows that the function has two real roots, one
between 23 and 22 and the other between 0 and
1. Can we find the exact values of the roots of the
function y 5 x2 1 2x 2 1? We know that the equa1
tion 0 5 x2 1 2x 2 1 cannot be solved by factoring
the right member over the set of integers.
x
O 1
However, we can follow a process similar to that
used to solve 0 5 x2 2 2. We need to write the right
y x2 2x 1
member as a trinomial that is the square of a binomial. This process is called completing the square.
Recall that (x 1 h)2 5 x2 1 2hx 1 h2. To write a trinomial that is a perfect square using the terms x2 1 2x, we need to let 2h 5 2. Therefore, h 5 1
and h2 5 1. The perfect square trinomial is x2 1 2x 1 1, which is the square of
(x 1 1).
(1) Add 1 to both sides of the equation to
isolate the terms in x:
0 5 x2 1 2x 2 1
1 5 x2 1 2x
(2) Add the number needed to complete
the square to both sides of the equation:
2 5 x2 1 2x 1 1
2 5 (x 1 1)2
A 12 coefficient of x B
2
2
5 A 12 (2) B 5 12 5 1
(3) Write the square root of both sides
of the equation:
(4) Solve for x:
6 !2 5 x 1 1
21 6 !2 5 x
The roots of y = x 1 2x 2 1 are 21 2 !2 and 21 1 !2.
2
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We can use a calculator to approximate these roots to the nearest
hundredth.
ENTER:
(-)
1 2nd
¯
ENTER:
2 ENTER
DISPLAY:
(-)
1 2nd
¯
2 ENTER
DISPLAY:
-1-√(2
-1+√(2
-2.414213562
.4142135624
To the nearest hundredth, the roots are 22.41 and 0.41. One root is between
23 and 22 and the other is between 0 and 1, as shown on the graph.
The process of completing the square can be illustrated geometrically. Let
us draw the quadratic expression x2 1 2x as a square with sides of length x and
a rectangle with dimensions x 3 2. Cut the rectangle in half and attach the two
parts to the sides of the square. The result is a larger square with a missing corner. The factor
A 12 coefficient of x B
2
or 12 is the area of this missing corner. This
is the origin of the expression “completing the square.”
(x 1)2
x2
2x
⇒
x
x
2
1
12
x
x
1
Procedure
To solve a quadratic equation of the form ax2 1 bx 1 c 5 0 where
a 5 1 by completing the square:
1. Isolate the terms in x on one side of the equation.
2
2. Add the square of one-half the coefficient of x or A 12 b B to both sides of the
equation.
3. Write the square root of both sides of the resulting equation and solve for x.
Note: The procedure outlined above only works when a 5 1. If the coefficient a
is not equal to 1, divide each term by a, and then proceed with completing the
square using the procedure outlined above.
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Not every second-degree polynomial
function has real roots. Note that the graph of
y 5 x2 2 6x 1 10 does not intersect the x-axis
and therefore has no real roots. Can we find
roots by completing the square?
189
y
1
O
y x2 6x 10
1
x
2
2
Use A 12 coefficient of x B 5 A 12 (26) B 5 (23) 2 5 9 to complete the square.
0 5 x2 2 6x 1 10
210 5 x2 2 6x
Isolate the terms in x.
2
210 1 9 5 x 2 6x 1 9
2
21 5 (x 2 3)
6 !21 5 x 2 3
Add the square of one-half the
coefficient of x to complete square.
Take the square root of both sides.
3 6 !21 5 x
Solve for x.
Since there is no real number that is the square root of 21, this function has
no real zeros. We will study a set of numbers that includes the square roots of
negative numbers later in this chapter.
EXAMPLE 1
Find the exact values of the zeros of the function f(x) 5 2x2 2 4x 1 2.
Solution
How to Proceed
0 5 2x2 2 4x 1 2
(1) Let f(x) 5 0:
(2) Isolate the terms in x:
2 5 x2 1 4x
(3) Multiply by 21:
(4) Complete the square by adding
A 12 coefficient of x B
22 5 2x2 2 4x
2
2
5 A 12 (4) B 5 22 5 4
to both sides of the equation:
(5) Take the square root of each side of the
equation:
(6) Solve for x:
Answer The zeros are (22 1 !6) and (22 2 !6) .
4 1 2 5 x2 1 4x 1 4
6 5 x2 1 4x 1 4
6 5 (x 1 2)2
6 !6 5 x 1 2
22 6 !6 5 x
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EXAMPLE 2
Find the roots of the function y 5 2x2 2 x 2 1.
How to Proceed
Solution
(1) Let y 5 0:
0 5 2x2 2 x 2 1
(2) Isolate the terms in x:
1 5 2x2 2 x
(3) Divide both sides of the equation by
the coefficient of x2:
1
2
1
2
1
16
(4) Complete the square by adding
A 12 coefficient of x B
2
1
16
1
5 A 12 ? 212 B 5 16
2
1
5 x2 2 12x
5 x2 2 12x 1 A 214 B
2
2
8
1 16
5 A x 2 14 B
9
16
to both sides of the equation:
5 A x 2 14 B
2
9
6#16
5 x 2 14
(5) Take the square root of both sides of
the equation:
1
4
(6) Solve for x:
6 34 5 x
The roots of the function are:
1
4
2 34 5 224 5 212
1
4
and
1 34 5 44 5 1
Answer 212 and 1
Note that since the roots are rational, the equation could have been solved
by factoring over the set of integers.
0 5 2x2 2 x 2 1
0 5 (2x 1 1)(x 2 1)
2x 1 1 5 0
x2150
2x 5 21
x51
x 5 212
Graphs of Quadratic Functions
and Completing the Square
A quadratic function y 5 a(x 2 h) 2 1 k can be graphed in terms of the basic
quadratic function y 5 x2. In particular, the graph of y 5 a(x 2 h) 2 1 k is the
graph of y 5 x2 shifted h units horizontally and k units vertically. The coefficient
a determines its orientation: when a . 0, the parabola opens upward; when
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Real Roots of a Quadratic Equation
a , 0, the parabola opens downward. The coefficient a also determines how narrow or wide the graph is compared to y 5 x2: when a . 1, the graph is narrower;
when a , 1, the graph is wider.
We can use the process of completing the square to help us graph quadratic
functions of the form y 5 ax2 1 bx 1 c, as the following examples demonstrate:
EXAMPLE 3
Graph f(x) 5 x2 1 4x 1 9.
Solution The square of one-half the coefficient of x is
2
A 12 ? 4 B or 4.
y
f(x)
Add and subtract 4 to keep the function equivalent to the original and complete the square.
f(x) 5 x2 1 4x 1 9
5 x2 1 4x 1 4 2 4 1 9
5 (x2 1 4x 1 4) 1 5
y x2
5 (x 1 2) 2 1 5
1
2
The graph of f(x) is the graph of y 5 x shifted 2
units to the left and 5 units up, as shown on the
right. Note that this function has no real roots.
1 O
x
EXAMPLE 4
Graph f(x) 5 2x2 1 4x 1 1.
Solution
How to Proceed
(1) Factor 2 from the first two terms:
y
f(x) 5 2x2 1 4x 1 1
5 2(x2 1 2x) 1 1
(2) Add and subtract
1
y x2
O 1
f(x)
x
A 12 coefficient of x B
2
2
5 A 12 ? 2 B 5 1
to keep the function equivalent to the
original:
(3) Complete the square:
5 2(x2 1 2x 1 1 2 1) 1 1
5 2(x2 1 2x 1 1) 2 2 1 1
5 2(x2 1 2x 1 1) 2 1
5 2(x 1 1) 2 2 1
The graph of f(x) is the graph of y 5 x2 stretched vertically by a factor of 2,
and shifted 1 unit to the left and 1 unit down as shown on the left. Note that
this function has two real roots, one between 22 and 21 and the other
between 21 and 0.
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Exercises
Writing About Mathematics
1. To solve the equation given in step 2 of Example 2, Sarah multiplied each side of the equation by 8 and then added 1 to each side to complete the square. Show that 16x2 2 8x 2 8 is
the square of a binomial and will lead to the correct solution of 0 5 2x2 2 x 2 1.
2. Phillip said that the equation 0 5 x2 2 6x 1 1 can be solved by adding 8 to both sides of the
equation. Do you agree with Phillip? Explain why or why not.
Developing Skills
In 3–8, complete the square of the quadratic expression.
3. x2 1 6x
4. x2 2 8x
5. x2 2 2x
6. x2 2 12x
7. 2x2 2 4x
8. x2 2 3x
In 9–14: a. Sketch the graph of each function. b. From the graph, estimate the roots of the function
to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
9. f(x) 5 x2 2 6x 1 4
12. f(x) 5 x2 2 6x 1 6
10. f(x) 5 x2 2 2x 2 2
11. f(x) 5 x2 1 4x 1 2
13. f(x) 5 x2 2 2x 2 1
14. f(x) 5 x2 2 10x 1 18
In 9–26, solve each quadratic equation by completing the square. Express the answer in simplest
radical form.
15. x2 2 2x 2 2 5 0
16. x2 1 6x 1 4 5 0
17. x2 2 4x 1 1 5 0
18. x2 1 2x 2 5 5 0
19. x2 2 6x 1 2 5 0
20. x2 2 8x 1 4 5 0
21. 2x2 1 12x 1 3 5 0
22. 3x2 2 6x 2 1 5 0
23. 2x2 2 6x 1 3 5 0
26. 14x2 1 34x 2 32 5 0
27. a. Complete the square to find the roots of the equation x2 2 5x 1 1 5 0.
24. 4x2 2 20x 1 9 5 0
25. 12x2 1 x 2 3 5 0
b. Write, to the nearest tenth, a rational approximation for the roots.
In 28–33, without graphing the parabola, describe the translation, reflection, and/or scaling that must
be applied to y 5 x2 to obtain the graph of each given function.
28. f(x) 5 x2 2 12x 1 5
29. f(x) 5 x2 1 2x 2 2
30. f(x) 5 x2 2 6x 2 7
31. f(x) 5 x2 1 x 1 94
32. f(x) 5 2x2 1 x 1 2
33. f(x) 5 3x2 1 6x 1 3
34. Determine the coordinates of the vertex and the equation of the axis of symmetry of
f(x) 5 x2 1 8x 1 5 by writing the equation in the form f(x) 5 (x 2 h)2 1 k. Justify your
answer.
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193
Applying Skills
35. The length of a rectangle is 4 feet more than twice the width. The area of the rectangle is 38
square feet.
a. Find the dimensions of the rectangle in simplest radical form.
b. Show that the product of the length and width is equal to the area.
c. Write, to the nearest tenth, rational approximations for the length and width.
36. One base of a trapezoid is 8 feet longer than the other base and the height of the trapezoid
is equal to the length of the shorter base. The area of the trapezoid is 20 square feet.
a. Find the lengths of the bases and of the height of the trapezoid in simplest radical form.
b. Show that the area of the trapezoid is equal to one-half the height times the sum of the
lengths of the bases.
c. Write, to the nearest tenth, rational approximations for the lengths of the bases and for
the height of the trapezoid.
37. Steve and Alice realized that their ages are consecutive odd integers. The product of their
ages is 195. Steve is younger than Alice. Determine their ages by completing the square.
5-2 THE QUADRATIC FORMULA
We have solved equations by using the same steps to complete the square needed
to find the roots or zeros of a function of the form y 5 ax2 1 bx 1 c, that is, to solve
the quadratic equation 0 5 ax2 1 bx 1 c. We will apply those steps to the general
equation to find a formula that can be used to solve any quadratic equation.
How to Proceed
(1) Let y 5 0:
0 5 ax2 + bx + c
(2) Isolate the terms in x:
2c 5 ax2 1 bx
(3) Divide both sides of the equation by
the coefficient of x2:
2c
a
(4) Complete the square by adding
A 12 coefficient of x B
2
b2
5 A 12 ? ba B 5 4a
2
to both sides of the equation:
2
(5) Take the square root of each side of
the equation:
(6) Solve for x:
b2
4a2
b2
4a2
5 x2 1 bax
b
b2
2
1 2c
a 5 x 1 a x 1 4a2
b
b2
2
1 24ac
4a2 5 x 1 a x 1 4a2
b2 2 4ac
4a2
b2 2 4ac
6# 4a2
"b2 2 4ac
2b
2a 6
2a
2b 6 "b2 2 4ac
2a
b
5 A x 1 2a
B
b
5 x 1 2a
5x
5x
This result is called the quadratic formula.
b2 2 4ac
When a 0, the roots of ax2 1 bx 1 c 5 0 are x 5 2b 6 "2a
.
2
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EXAMPLE 1
Use the quadratic formula to find the roots of 2x2 2 4x 5 1.
How to Proceed
Solution
2x2 2 4x 5 1
(1) Write the equation in standard form:
2x2 2 4x 2 1 5 0
(2) Determine the values of a, b, and c:
(3) Substitute the values of a, b, and c in the
quadratic formula:
a 5 2, b 5 24, c 5 21
2
x 5 2b 6 "2ab 2 4ac
x5
(4) Perform the computation:
2(24) 6 " (24) 2 2 4(2)(21)
2(2)
1 8
x 5 4 6 !16
4
x 5 4 6 4!24
(5) Write the radical in simplest form:
!6
x 5 4 6 !4
4
x 5 4 6 42 !6
(6) Divide numerator and denominator by 2:
Answer
2 1 !6
2
x 5 2 62 !6
and 2 22 !6
EXAMPLE 2
Use the quadratic formula to show that the equation x2 1 x 1 2 5 0 has no real
roots.
Solution For the equation x2 1 x 1 2 5 0, a 5 1, b 5 1, and c 5 2.
2
21 6 "12 2 4(1)(2)
21 6 !1 2 8
21 6 !27
5
5
x 5 2b 6 "2ab 2 4ac 5
2
2
2(1)
There is no real number that is the square root of a negative number.
Therefore, !27 is not a real number and the equation has no real roots.
EXAMPLE 3
One leg of a right triangle is 1 centimeter shorter than the other leg and the
hypotenuse is 2 centimeters longer than the longer leg. What are lengths of the
sides of the triangle?
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195
Solution Let x 5 the length of the longer leg
x 2 1 5 the length of the shorter leg
x 1 2 5 the length of the hypotenuse
Use the Pythagorean Theorem to write an equation.
x2 1 (x 2 1)2 5 (x 1 2)2
x2 1 x2 2 2x 1 1 5 x2 1 4x 1 4
x2 2 6x 2 3 5 0
Use the quadratic formula to solve this equation: a 5 1, b 5 26, c 5 23.
2
x 5 2b 6 "2ab 2 4ac
x5
5
5
2(26) 6 " (26) 2 2 4(1)(23)
2(1)
6 6 !36 1 12
2
6 6 !48
!3
5 6 6 !16
5 6 6 24 !3
2
2
5 3 6 2 !3
The roots are 3 1 2 !3 and 3 2 2!3. Reject the negative root, 3 2 2 !3.
The length of the longer leg is 3 1 2 !3.
The length of the shorter leg is 3 1 2 !3 2 1 or 2 1 2!3.
The length of the hypotenuse is 3 1 2!3 1 2 or 5 1 2!3.
Check
A3 1 2!3B 2 1 A2 1 2 !3B 2 5 A5 1 2!3B 2
?
A9 1 12 !3 1 4 ? 3B 1 A4 1 8 !3 1 4 ? 3B 5 25 1 20!3 1 4 ? 3
?
37 1 20!3 5 37 1 20 !3 ✔
Exercises
Writing About Mathematics
1. Noah said that the solutions of Example 1, 2 12 !6 and 2 22 !6, could have been written as
2 1 !6
2
1
1
5 1 1 !6 and 2 22 !6 5 1 2 !6. Do you agree with Noah? Explain why or why
1
1
not.
2. Rita said that when a, b, and c are real numbers, the roots of ax2 1 bx + c 5 0 are real numbers only when b2 $ 4ac. Do you agree with Rita? Explain why or why not.
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Developing Skills
In 3–14, use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form.
3. x2 1 5x 1 4 5 0
4. x2 1 6x 2 7 5 0
5. x2 2 3x 1 1 5 0
6. x2 2 x 2 4 5 0
7. x2 1 5x 2 2 5 0
8. x2 2 8 5 0
9. x2 2 3x 5 0
10. x2 1 2x 5 4
11. 3x2 2 5x 1 2 5 0
12. 4x2 2 x 2 1 5 0
13. 5x 1 1 5 2x2
14. 2x2 5 x 1 4
15. x2 2 6x 1 3 5 0
16. 4x2 2 4x 5 11
17. 3x2 5 4x 1 2
18. a. Sketch the graph of y 5 x2 1 6x 1 2.
b. From the graph, estimate the roots of the function to the nearest tenth.
c. Use the quadratic formula to find the exact values of the roots of the function.
d. Express the roots of the function to the nearest tenth and compare these values to
your estimate from the graph.
Applying Skills
In 19–25, express each answer in simplest radical form. Check each answer.
19. The larger of two numbers is 5 more than twice the smaller. The square of the smaller is
equal to the larger. Find the numbers.
20. The length of a rectangle is 2 feet more than the width. The area of the rectangle is 2 square
feet. What are the dimensions of the rectangle?
21. The length of a rectangle is 4 centimeters more than the width. The measure of a diagonal is
10 centimeters. Find the dimensions of the rectangle.
22. The length of the base of a triangle is 6 feet more than the length of the altitude to the base.
The area of the triangle is 18 square feet. Find the length of the base and of the altitude to
the base.
23. The lengths of the bases of a trapezoid are x 1 10 and 3x 1 2 and the length of the altitude
is 2x. If the area of the trapezoid is 40, find the lengths of the bases and of the altitude.
24. The altitude, CD, to the hypotenuse, AB, of right triangle ABC
separates the hypotenuse into two segments, AD and DB. If
AD 5 DB 1 4 and CD 5 12 centimeters, find DB, AD, and AB.
Recall that the length of the altitude to the hypotenuse of a right
triangle is the mean proportional between the lengths of the
segments into which the hypotenuse is separated, that is,
AD
CD
CD
.
5 DB
A
DB 4
D
12 cm
C
B
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197
25. A parabola is symmetric under a line reflection. Each real root of the quadratic function
y 5 ax2 1 bx + c is the image of the other under a reflection in the axis of symmetry of the
parabola.
a. What are the coordinates of the points at which the parabola whose equation is
y 5 ax2 1 bx + c intersects the x-axis?
b. What are the coordinates of the midpoint of the points whose coordinates were found
in part a?
c. What is the equation of the axis of symmetry of the parabola y 5 ax2 1 bx + c?
d. The turning point of a parabola is on the axis of symmetry. What is the x-coordinate of
the turning point of the parabola y 5 ax2 1 bx + c?
26. Gravity on the moon is about one-sixth of gravity on Earth. An astronaut standing on a
tower 20 feet above the moon’s surface throws a ball upward with a velocity of 30 feet
per second. The height of the ball at any time t (in seconds) is h(t) 5 22.67t2 1 30t 1 20.
To the nearest tenth of a second, how long will it take for the ball to hit the ground?
27. Based on data from a local college, a statistician determined that the average student’s
grade point average (GPA) at this college is a function of the number of hours, h, he or
she studies each week. The grade point average can be estimated by the function
G(h) 5 0.006h2 1 0.02h 1 1.2 for 0 # h # 20.
a. To the nearest tenth, what is the average GPA of a student who does no studying?
b. To the nearest tenth, what is the average GPA of a student who studies 12 hours per
week?
c. To the nearest tenth, how many hours of studying are required for the average student to
achieve a 3.2 GPA?
28. Follow the steps given in the chapter opener to find a root of the equation x2 1 bax 1 ac 5 0.
Compare your solution with the quadratic formula.
Hands-On Activity: Alternate Derivation of the Quadratic Formula
As you may have noticed, the derivation of the quadratic formula on page 193 uses fractions
throughout (in particular, starting with step 3). We can use an alternate derivation of the quadratic
equation that uses fractions only in the last step. This alternate derivation uses the following perfect
square trinomial:
4a2x2 1 4abx 1 b2 5 4a2x2 1 2abx 1 2abx + b2
5 2ax(2ax + b) 1 b(2ax + b)
5 (2ax + b)(2ax + b)
5 (2ax + b)2
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To write ax2 1 bx, the terms in x of the general quadratic equation, as this perfect square trinomial, we must multiply by 4a and add b2. The steps for the derivation of the quadratic formula using
this perfect square trinomial are shown below.
How to Proceed
(1) Isolate the terms in x:
(2) Multiply both sides of the equation by 4a:
(3) Add b2 to both sides of the equation to complete
the square:
(4) Now solve for x.
ax2 1 bx + c 5 0
ax2 1 bx 5 2c
2 2
4a x 1 4abx 5 24ac
2 2
4a x 1 4abx 1 b2 5 b2 2 4ac
(2ax 1 b)2 5 b2 2 4ac
Does this procedure lead to the quadratic formula?
5-3 THE DISCRIMINANT
When a quadratic equation ax2 1 bx 1 c 5 0 has rational numbers as the values of a, b, and c, the value of b2 2 4ac, or the discriminant, determines the
nature of the roots. Consider the following equations.
CASE 1
The discriminant b2 2 4ac is a positive perfect square.
2x2 1 3x 2 2 5 0 where a 5 2, b 5 3, and c 5 22.
2
x 5 2b 6 "2ab 2 4ac
x5
y
23 6 "32 2 4(2)(22)
2(2)
9 1 16
5 23 6 "
5 23 64 "25 5 23 46 5
4
The roots are 23 41 5 5 24 5 12 and
5 28
4 5 22.
The roots are unequal rational numbers,
and the graph of the corresponding quadratic function has two x-intercepts.
1
O
x
1
23 2 5
4
y 2x2 3x 2
CASE 2 The discriminant b2 2 4ac is a positive
number that is not a perfect square.
y
3x2 1 x 2 1 5 0 where a 5 3, b 5 1, and c 5 21.
2
x 5 2b 6 "2ab 2 4ac
x5
21 6 "12 2 4(3)(21)
2(3)
1 12
5 21 6 !1
5 21 66 !13
6
The roots are 21 16 !13 and 21 26 !13.
1
O
1
x
y 3x2 x 1
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199
The roots are unequal irrational numbers, and the graph of the corresponding quadratic function has two x-intercepts.
CASE 3
The discriminant b2 2 4ac is 0.
9x2 2 6x 1 1 5 0 where a 5 9, b 5 26, and c 5 1.
y
x 5 2b 6 "2a 2 4ac
b2
x5
2(26) 6 " (26) 2 2 4(9)(1)
2(9)
36 2 36
6 0
5 6 6 "18
5 6 618"0 5 6 18
2 0
1 0
5 13 and 6 18
5 13.
The roots are 6 18
The roots are equal rational numbers, and the
graph of the corresponding quadratic function has
one x-intercept. The root is a double root.
CASE 4
1
O
1
x
y 9x2 6x 1
The discriminant b2 2 4ac is a negative number.
2x2 1 3x 1 4 5 0 where a 5 2, b 5 3, and
c 5 4.
y
2
x 5 2b 6 "2ab 2 4ac
x5
23 6 "32 2 4(2)(4)
2(2)
2 32
5 23 6 !9
5 23 6 4!223
4
This equation has no real roots because
there is no real number equal to !223, and the
graph of the corresponding quadratic function
has no x-intercepts.
y 2x2 3x 4
1
O
1
x
In each case, when a, b, and c are rational numbers, the value of b2 2 4ac
determined the nature of the roots and the x-intercepts. From the four cases
given above, we can make the following observations.
Let ax2 1 bx 1 c 5 0 be a quadratic equation and a, b, and c be rational
numbers with a 0:
The number
of x-intercepts
of the function is:
When the discriminant
b2 2 4ac is:
The roots of the
equation are:
. 0 and a perfect square
real, rational, and unequal
2
. 0 and not a perfect square
real, irrational, and unequal
2
50
real, rational, and equal
1
,0
not real numbers
0
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EXAMPLE 1
Without solving the equation, find the nature of the roots of 3x2 1 2x 5 1.
Solution Write the equation in standard form: 3x2 1 2x 2 1 5 0.
Evaluate the discriminant of the equation when a 5 3, b 5 2, c 5 21.
b2 2 4ac 5 22 2 4(3)(21) 5 4 1 12 5 16
Since a, b, and c are rational and the discriminant is greater than 0 and a perfect square, the roots are real, rational, and unequal.
EXAMPLE 2
Show that it is impossible to draw a rectangle whose area is 20 square feet and
whose perimeter is 10 feet.
Solution Let l and w be the length and width of the rectangle.
How to Proceed
(1) Use the formula for perimeter:
(2) Solve for l in terms of w:
2l 1 2w 5 10
l1w55
l552w
(3) Use the formula for area:
(4) Substitute 5 2 w for l. Write the
resulting quadratic equation in
standard form:
lw 5 20
(5 2 w)(w) 5 20
5w 2 w2 5 20
2w2 1 5w 2 20 5 0
w2 2 5w 1 20 5 0
(5) Test the discriminant:
b2 2 4ac 5 (25)2 2 4(1)(20)
5 25 2 80
5 255
Since the discriminant is negative, the quadratic equation has no real roots.
There are no real numbers that can be the dimensions of this rectangle.
EXAMPLE 3
A lamp manufacturer earns a weekly profit of P dollars according to the
function P(x) 5 20.3x2 1 50x 2 170 where x is the number of lamps sold. When
P(x) 5 0, the roots of the equation represent the level of sales for which the
manufacturer will break even.
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a. Interpret the meaning of the discriminant value of 2,296.
b. Use the discriminant to determine the number of break even points.
c. Use the discriminant to determine if there is a level of sales for which the
weekly profit would be $3,000.
Solution a. Since the discriminant is greater than 0 and not a perfect square, the
equation has two roots or break even points that are irrational and
unequal. Answer
b. 2 (see part a) Answer
c. We want to know if the equation
3,000 5 20.3x2 1 50x 2 170
has at least one real root. Find the discriminant of this equation. The equation in standard form is 0 5 20.3x2 1 50x 2 3,170.
b2 2 ac 5 502 2 4(20.3)(23,170) 5 21,304
The discriminant is negative, so there are no real roots. The company cannot
achieve a weekly profit of $3,000. Answer
Exercises
Writing About Mathematics
1. a. What is the discriminant of the equation x2 1 A !5B x 2 1 5 0?
b. Find the roots of the equation x2 1 A !5B x 2 1 5 0.
c. Do the rules for the relationship between the discriminant and the roots of the equation given in this chapter apply to this equation? Explain why or why not.
2. Christina said that when a, b, and c are rational numbers and a and c have opposite signs,
the quadratic equation ax2 1 bx + c 5 0 must have real roots. Do you agree with Christina?
Explain why or why not.
Developing Skills
In 3–8, each graph represents a quadratic function. Determine if the discriminant of the quadratic
function is greater than 0, equal to 0, or less than 0.
3.
4.
y
5.
y
y
O
x
O
x
O
x
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Quadratic Functions and Complex Numbers
6.
7.
y
8.
y
y O
x
O
x
O
x
In 9–14: a. For each given value of the discriminant of a quadratic equation with rational coefficients,
determine if the roots of the quadratic equation are (1) rational and unequal, (2) rational and equal,
(3) irrational and unequal, or (4) not real numbers. b. Use your answer to part a to determine the
number of x-intercepts of the graph of the corresponding quadratic function.
9. 9
10. 5
11. 0
12. 12
13. 24
14. 49
In 15–23: a. Find the value of the discriminant and determine if the roots of the quadratic equation
are (1) rational and unequal, (2) rational and equal, (3) irrational and unequal, or (4) not real numbers. b. Use any method to find the real roots of the equation if they exist.
15. x2 2 12x 1 36 5 0
16. 2x2 1 7x 5 0
17. x2 1 3x 1 1 5 0
18. 2x2 2 8 5 0
19. 4x2 2 x 5 1
20. 3x2 5 5x 2 3
21. 4x 2 1 5 4x2
22. 2x2 2 3x 2 5 5 0
23. 3x 2 x2 5 5
24. When b2 2 4ac 5 0, is ax2 1 bx 1 c a perfect square trinomial or a constant times a perfect
square trinomial? Explain why or why not.
25. Find a value of c such that the roots of x2 1 2x 1 c 5 0 are:
a. equal and rational.
b. unequal and rational.
c. unequal and irrational.
d. not real numbers.
26. Find a value of b such that the roots of x2 1 bx 1 4 5 0 are:
a. equal and rational.
b. unequal and rational.
c. unequal and irrational.
d. not real numbers.
Applying Skills
27. Lauren wants to fence off a rectangular flower bed with a perimeter of 30 yards and a diagonal length of 8 yards. Use the discriminant to determine if her fence can be constructed. If
possible, determine the dimensions of the rectangle.
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28. An open box is to be constructed as shown in the figure
on the right. Is it possible to construct a box with a volume of 25 cubic feet? Use the discriminant to explain
your answer.
203
4 ft
x
5x
29. The height, in feet, to which of a golf ball rises when shot upward from ground level is
described by the function h(t) 5 216t2 1 48t where t is the time elapsed in seconds. Use
the discriminant to determine if the golf ball will reach a height of 32 feet.
30. The profit function for a company that manufactures cameras is P(x) 5 2x2 1 350x 2 15,000.
Under present conditions, can the company achieve a profit of $20,000? Use the discriminant to explain your answer.
5-4 THE COMPLEX NUMBERS
The Set of Imaginary Numbers
We have seen that when we solve quadratic equations with rational coefficients,
some equations have real, rational roots, some have real, irrational roots, and
some do not have real roots.
For example:
x2 2 9 5 0
x2 2 8 5 0
x2 5 9
x2 5 8
x 5 6 !9
x 5 63
x 5 6 !8
x 5 6 !4!2
x 5 62 !2
x2 1 9 5 0
x2 5 29
x 5 6 !29
x 5 6 !9!21
x 5 63 !21
The equations x2 2 9 5 0 and x2 2 8 5 0 have real roots, but the equation
x2 1 9 5 0 does not have real roots.
A number is an abstract idea to which we assign a meaning and a symbol.
We can form a new set of numbers that includes !21, to which we will assign
the symbol i. We call i the unit of the set of imaginary numbers.
The numbers 13 !21 and 23 !21 that are the roots of the equation
x2 1 9 5 0 can be written as 3i and 23i.
DEFINITION
A number of the form a !21 5 ai is a pure imaginary number where a is a
non-zero real number.
The idea of the square root of 21 as a number was not accepted by many
early mathematicians. Rafael Bombelli (1526–1572) introduced the terms plus
of minus and minus of minus to designate these numbers. For instance, he
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designated 2 1 3i as 2 plus of minus 3 and 2 2 3i as 2 minus of minus 3. The term
imaginary was first used in 1637 by René Descartes, who used it in an almost
derogatory sense, implying that somehow imaginary numbers had no significance in mathematics. However, the algebra of imaginary numbers was later
developed and became an invaluable tool in real-world applications such as
alternating current.
Some elements of the set of pure imaginary numbers are:
!225 5 !25!21 5 5i
2!216 5 2!4!21 5 22i
!210 5 !10!21 5 !21!10 5 i !10
2!232 5 2!16!2!21 5 24 !21 !2 5 24i !2
Note the order in which the factors of i !10 and 24i !2 are written. The factor that is a radical is written last to avoid confusion with !10i and 24!2i.
EXAMPLE 1
Simplify: a. !27
b. !212
Solution Write each imaginary number in terms of i and simplify.
a. !27 5 !7!21 5 i !7
b. !212 5 !4!3!21 5 2i !3
Powers of i
The unit element of the set of pure imaginary
numbers is i 5 !21. Since !b is one of the two
equal factors of b, then i 5 !21 is one of the
two equal factors of 21 and i i 5 21 or i2 5 21.
The box at the right shows the first four powers
of i.
In general, for any integer n:
i 5 !21
i 2 5 21
i 3 5 i 2 i 5 21i 5 2i
i 4 5 i 2 i 2 5 21(21) 5 1
i4n 5 (i4)n 5 1n 5 1
i4n11 5 (i4)n(i1) 5 1n(i) 5 1(i) 5 i
i4n12 5 (i4)n(i2) 5 1n(21) 5 1(21) 5 21
i4n13 5 (i4)n(i3) 5 1n(2i) 5 1(2i) 5 2i
For any positive integer k, ik is equal to i, 21, 2i, or 1. We can use this relationship to evaluate any power of i.
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205
EXAMPLE 2
Evaluate:
Answers
a. i14
5 i4(3)12 5 (i4)3 (i2) 5 (1)3(21) 5 1(21) 5 21
b. i7
5 i7 5 i413 5 i4 ? i3 5 (1)(2i) 5 2i
Arithmetic of Imaginary Numbers
The rule for multiplying radicals, !a ? !b 5 !ab, is true if and only if a and b
are non-negative numbers. When a , 0 or b , 0, !a ? !b 2 !ab. For example, it is incorrect to write: !24 3 !29 5 !36 5 6, but it is correct to write:
!24 3 !29 5 !4!21 3 !9!21 5 2i 3 3i 5 6i2 5 6(21) 5 26
The distributive property of multiplication over addition is true for imaginary numbers.
!24 1 !29
!28 1 !250
5 !4!21 1 !9!21
5 !4!21!2 1 !25!21!2
5 (2 1 3)i
5 (2 1 5) Ai !2B
5 2i 1 3i
5 5i
5 2i !2 1 5i !2
5 7i !2
EXAMPLE 3
Find the sum of !225 1 !249.
Solution Write each imaginary number in terms of i and add similar terms.
!225 1 !249 5 !25!21 1 !49!21 5 5i 1 7i 5 12i Answer
The Set of Complex Numbers
The set of pure imaginary numbers is not closed under multiplication because
the powers of i include both real numbers and imaginary numbers. For example,
i i 5 21. We want to define a set of numbers that includes both real numbers
and pure imaginary numbers and that satisfies all the properties of the real
numbers. This new set of numbers is called the set of complex numbers and is
defined as the set of numbers of the form a + bi where a and b are real numbers.
When a 5 0, a 1 bi 5 0 1 bi 5 bi, a pure imaginary number. When b 5 0,
a 1 bi 5 a 1 0i 5 a, a real number. Therefore, both the set of pure imaginary
numbers and the set of real numbers are contained in the set of complex numbers. A complex number that is not a real number is called an imaginary number.
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DEFINITION
A complex number is a number of the form
a 1 bi
where a and b are real numbers and i 5 !21.
Examples of complex numbers are:
2, 24, 13, !2, p, 4i, 25i, 14i, i !5, pi,
2 2 3i, 27 1 i, 8 1 3i, 0 2 i, 5 1 0i
Complex Numbers and the Graphing Calculator
The TI family of graphing calculators can use complex numbers. To change your
calculator to work with complex numbers, press MODE , scroll down to the row
that lists REAL, select a 1 bi, and then press ENTER . Your calculator will now
evaluate complex numbers. For example, with your calculator in a 1 bi mode,
write 6 1 !29 as a complex number in terms of i.
ENTER: 6
DISPLAY:
¯ 29
2nd
)
ENTER
6+√(-9)
6+3i
EXAMPLE 4
Evaluate each expression using a calculator:
a. !225 2 !216
b. 2 1 !29 1 5 !24
Calculator a. Press 2nd ¯ 225
Solution
ENTER .
216 )
b. Press 2 2nd
Answers a. i
b. 2 1 13i
¯ 29
2nd
¯ 24
)
)
ENTER .
)
¯
2nd
5
√(-25)-√(-16)
i
2+√(-9)+5√(-4)
2+13i
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The Complex Numbers
<
What about powers of i? A calculator is not
an efficient tool for evaluating an expression such
as bin for large values of n. Large complex powers
introduce rounding errors in the answer as shown
in the calculator screen on the right.
207
i 7
-3e-13-i
Graphing Complex Numbers
We are familiar with the one-to-one correspondence between the set of real
numbers and the points on the number line. The complex numbers can be put
in a one-to-one correspondence with the points of a plane. This correspondence
makes it possible for a physical quantity, such as an alternating current that has
both amplitude and direction, to be represented by a complex number.
On graph paper, draw a horizontal real
number line. Through 0 on the real number
4i
line, draw a vertical line. This vertical line is
the pure imaginary number line. On this line,
3i
mark points equidistant from one another
2i
such that the distance between them is equal
i
O
to the unit distance on the real number line.
4 3 2 1
1 2 3 4
Label the points above 0 as i, 2i, 3i, 4i, and
i
so on and the numbers below 0 as 2i, 22i,
2i
23i, 24i, and so on.
3i
To find the point that corresponds to the
4i
complex number 4 2 3i, move to 4 on the real
number line and then, parallel to the imaginary number line, move down 3 units to a
point on the same horizontal line as 3i.
Hands-On Activity
On the complex plane, locate the points that correspond to 4 1 2i and 2 2 5i.
Draw a line segment from 0 to each of these points. Now draw a parallelogram
with these two line segments as adjacent sides. What is the complex number that
corresponds to the vertex of the parallelogram opposite 0? How is this complex
number related to 4 1 2i and 2 2 5i? Repeat the procedure given above for
each of the following pairs of complex numbers.
1. 2 1 4i, 3 1 i
2. –2 1 5i, 25 1 2i
3. 4 2 2i, 2 2 2i
4. –4 2 2i, 3 2 5i
5. 1 1 4i, 1 2 4i
6. 25 1 3i, 25 2 3i
7. 23 1 0i, 0 2 4i
8. 4 1 0i, 24 1 3i
9. 4, 2i
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Exercises
Writing About Mathematics
1. Pete said that !22 3 !28 5 !16 5 4. Do you agree with Pete? Explain why or why not.
2. Ethan said that the square of any pure imaginary number is a negative real number. Do you
agree with Ethan? Justify your answer.
Developing Skills
In 3–18, write each number in terms of i.
3. !24
7. 2!2121
11. 5 !227
15. 5 1 !25
4. !281
8. !28
12. 212 !280
16. 1 1 !23
5. !29
6. 2!236
9. !212
10. 2!272
14. !2500
13. 2!251
17. 24 2 !224
In 19–34, write each sum or difference in terms of i.
18. 23 1 2 !29
19. !2100 1 !281
20. !225 2 !24
23. !212 1 !227 2 !275
24. 2 !25 1 !2125
21. !2144 1 !21
22. !249 2 !216
25. !4 1 !24 1 !236 2 !36
26. !50 1 !22 1 !200 1 !250
27. !4 1 !232 1 !28
28. 3 1 !228 2 7 2 !27
29. 23 2 !210 1 2 2 !290
30. 22 1 !216 1 7 2 !249
31. 3 1 !236 2 6 1 !21
32. 17 1 #218 1 27 2 #212
34. !0.2025 1 !20.09
33. 212 1 #223 2 12 1 #224
9
In 35–43, write each number in simplest form.
35. 3i 1 2i 3
36. 5i 2 1 2i 4
37. i 2 5i 3
38. 2i 5 1 7i 7
39. i + i 3 1 i 5
40. 4i 1 5i 8 1 6i 3 1 2i 4
41. i 2 1 i 12 1 i 8
42. i + i 2 1 i 3 1 i 4
43. i 57
In 44–51, locate the point that corresponds to each of the given complex numbers.
44. 2 1 4i
48.
1
2
1 4i
45. –2 1 5i
46. 4 2 2i
47. –4 2 2i
49. 0 1 3i
50. 23 1 0i
51. 0 1 0i
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209
Applying Skills
52. Imaginary numbers are often used in electrical engineering. The impedance Z (measured in
ohms) of a circuit measures the resistance of a circuit to alternating current (AC) electricity.
For two AC circuits connected in series, the total impedance is the sum Z1 1 Z2 of the individual impedances. Find the total impedance if Z1 5 5i ohms and Z2 5 8i ohms.
ZZ
1 2
53. In certain circuits, the total impedance ZT is given by the formula ZT 5 Z 1
Z2. Find ZT
1
when Z1 5 23i and Z2 5 4i.
Hands-On Activity: Multiplying Complex Numbers
Multiplication by i
In the complex plane, the image of 1 under a rotation of 90° about the origin is i, which is equivalent to multiplying 1 by i. Now rotate i by 90° about the origin. The image of i is 21, which models
the product i i 5 21. This result is true for any complex number. That is, multiplying any complex
number a 1 bi by i is equivalent to rotating that number by 90° about the origin. Use this graphical
representation of multiplication by i to evaluate the following products:
1. (2 1 3i) i
2. 12i i
4. 25i i
5. 210 i
3. A 12 1 32i B ? i
6. (23 2 2i) i
Multiplication by a real number
In the complex plane, the image of any number a under a dilation of k is ak, which is equivalent to
multiplying a by k. The image any imaginary number ai under a dilation of k is aki, which is equivalent to multiplying ai by k. This result is true for any complex number. That is, multiplying any complex number a 1 bi by a real number k is equivalent to a dilation of k. Use this graphical
representation of multiplication by a real number to evaluate the following products:
1. (2 1 4i) 4
2. (25 2 5i) 12
4. 2i 2
5. 24i 12
3. A 34 1 2i B ? 4
6. (23 2 2i) 3i 5 (23 2 2i) 3 i
5-5 OPERATIONS WITH COMPLEX NUMBERS
Like the real numbers, the complex numbers are closed, commutative, and associative under addition and multiplication, and multiplication is distributive over
addition.
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Adding Complex Numbers
We can add complex numbers using the commutative, associative, and distributive properties that we use to add real numbers. For example:
(4 1 3i) 1 (2 1 5i) 5 4 1 (3i 1 2) 1 5i
Associative Property
5 4 1 (2 1 3i) 1 5i
Commutative Property
5 (4 1 2) 1 (3i 1 5i)
Associative Property
5 (4 1 2) 1 (3 1 5)i
Distributive Property
5 6 1 8i
Substitution Property
When we know that each of these steps leads to a correct sum, we can write:
(a + bi) 1 (c + di) 5 (a + c) 1 (b + d)i
Here are some other examples of the addition of complex numbers:
(8 2 i) 1 (3 2 7i) 5 (8 1 3) 1 (21 2 7)i 5 11 2 8i
(3 1 2i) 1 (1 2 2i) 5 (3 1 1) 1 (2 2 2)i 5 4 1 0i 5 4
(29 1 i) 1 (9 1 6i) 5 (29 1 9) 1 (1 1 6)i 5 0 1 7i 5 7i
Note that the sum of two complex numbers is always a complex number.
That sum may be an element of one of the subsets of the complex numbers, that
is, a real number or a pure imaginary number.
Since for any complex number a 1 bi:
(a 1 bi) 1 (0 1 0i) 5 (0 1 0i) 1 (a 1 bi) 5 (a 1 bi)
the real number (0 1 0i) or 0 is the additive identity.
The additive identity of the complex numbers is the real number (0 1 0i)
or 0.
Similarly, since for any complex number a 1 bi:
(a + bi) 1 (2a 2 bi) 5 [a 1 (2a)] 1 [b 1 (2b)]i 5 0 1 0i 5 0
the additive inverse of any complex number (a 1 bi) is (2a 2 bi).
The additive inverse of any complex number (a 1 bi) is (2a 2 bi).
On some calculators, the key for i is the 2nd function of the decimal point
key. When the calculator is in a 1 bi mode, we can add complex numbers.
ENTER: 4
DISPLAY:
2 2nd
i
4+2i+3-5i
7-3i
3 5 2nd
i
ENTER
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211
EXAMPLE 1
Express the sum in a + bi form: (23 1 7i) 1 (25 1 2i)
Solution (23 1 7i) 1 (25 1 2i) 5 (23 2 5) 1 (7 1 2)i = 28 1 9i
i
Calculator ENTER: ( 23 7 2nd
Solution
)
ENTER
i
2 2nd
DISPLAY:
(-3+7i)+(-5+2i)
)
(
25 -8+9i
Answer 28 1 9i
Subtracting Complex Numbers
In the set of real numbers, a 2 b 5 a 1 (2b), that is, a 2 b is the sum of a and
the additive inverse of b. We can apply the same rule to the subtraction of complex numbers.
EXAMPLE 2
Subtract:
Answers
a. (5 1 3i) 2 (2 1 8i)
5 (5 1 3i) 1 (22 2 8i) 5 (5 2 2) 1 (3 2 8)i
5 3 2 5i
b. (7 1 4i) 2 (21 1 3i)
5 (7 1 4i) 1 (1 2 3i) 5 (7 1 1) 1 (4 2 3)i
5 8 1 1i 5 8 1 i
c. (6 1 i) 2 (3 1 i)
5 (6 1 i) 1 (23 2 i) 5 (6 2 3) 1 (1 2 1)i
5 3 1 0i 5 3
d. (1 1 2i) 2 (1 2 9i)
5 (1 1 2i) 1 (21 1 9i) 5 (1 2 1) 1 (2 1 9)i
5 0 1 11i 5 11i
Multiplying Complex Numbers
We can multiply complex numbers using the same principles that we use to
multiply binomials. When simplifying each of the following products, recall that
i(i) 5 i2 5 21.
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EXAMPLE 3
Multiply:
a. (5 1 2i)(3 1 4i)
b. (3 2 5i)(2 1 i)
c. (9 1 3i)(9 2 3i)
d. (2 1 6i)2
Solution a.
(5 1 2i)(3 1 4i)
b.
5 5(3 1 4i) 1 2i(3 1 4i)
5 3(2 1 i) 2 5i(2 1 i)
2
c.
(3 2 5i)(2 1 i)
5 15 1 20i 1 6i 1 8i
5 6 1 3i – 10i 2 5i2
5 15 1 26i 1 8(21)
5 6 2 7i 2 5(21)
5 15 1 26i 2 8
5 6 2 7i 1 5
5 7 1 26i Answer
5 11 2 7i Answer
(9 1 3i)(9 2 3i)
d.
(2 1 6i) 2
5 9(9 2 3i) 1 3i(9 2 3i)
5 (2 1 6i)(2 1 6i)
5 81 2 27i 1 27i 2 9i2
5 2(2 1 6i) 1 6i(2 1 6i)
5 81 1 0i 2 9(21)
5 4 1 12i 1 12i 1 36i2
5 81 1 9
5 4 1 24i 1 36(21)
5 90 Answer
5 232 1 24i Answer
In part c of Example 3, the product is a real number.
Since for any complex number a 1 bi:
(a 1 bi)(1 1 0i) 5 a(1 1 0i) 1 bi(1 1 0i) 5 a 1 0i 1 bi 1 0i 5 a 1 bi
the real number 1 1 0i or 1 is the multiplicative identity.
The multiplicative identity of the complex numbers is the real number 1 1 0i
or 1.
Does every non-zero complex number have a multiplicative inverse? In the
set of rational numbers, the multiplicative inverse of a is a1. We can say that in
the set of complex numbers, the multiplicative inverse of a 1 bi is a 11 bi. To
express a 11 bi in the form of a complex number, multiply numerator and
denominator by the complex conjugate of a 1 bi, a 2 bi. The reason for doing
so is that the product of a complex number and its conjugate is a real number.
For example,
(1 1 2i)(1 2 2i) 5 1(1 2 2i) 1 2i(1 2 2i)
5 1 2 2i 1 2i 2 4i2
5 1 2 4(21)
55
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Operations with Complex Numbers
213
In general,
(a 1 bi)(a 2 bi) 5 a(a 2 bi) 1 bi(a 2 bi)
5 a2 2 abi 1 abi 2 b2i2
5 a2 2 b2(21)
5 a2 1 b2
or
(a 1 bi)(a 2 bi) 5 a2 1 b2
The product of a complex number a + bi and its conjugate a 2 bi is a real
number, a2 1 b2. Thus, if we the multiply numerator and denominator of a 11 bi
by the complex conjugate of the denominator, we will have an equivalent fraction with a denominator that is a real number.
1
a 1 bi
bi
a 2 bi
a
b
? aa 2
2 bi 5 a2 1 b2 5 a2 1 b2 2 a2 1 b2i
For example, the complex conjugate of 2 2 4i is 2 1 4i and the multiplicative inverse of 2 2 4i is
1
2 2 4i
4i
2 1 4i
4i
2
5 2 21 4i ? 22 1
2 4i 5 22 1 42 5 22 1 42 1 22 1 42
2
4
1 20
i
5 20
1
1 15i
5 10
1
We can check that 10
1 15i is the multiplicative inverse of 2 2 4i by multi1
plying the two numbers together. If 10
1 15i is indeed the multiplicative inverse,
then their product will be 1.
1
(2 2 4i) A 10
1 15i B 5 1
?
1
1
2 A 10
1 15i B 2 4i A 10
1 15i B 5 1
?
1
2 A 10
B 1 2 A 15i B 2 4i A 101 B 2 4i A 15i B 5 1
?
2
10
1
5
?
4
1 25i 2 10
i 2 45i2 5 1
?
1 25i 2 25i 2 45 (21) 5 1
4
1
5 1 551✔
For any non-zero complex number a 1 bi, its multiplicative inverse is
1
a 1 bi
or
a
a2 1 b2
b
2 a2 1
b2 i
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Quadratic Functions and Complex Numbers
EXAMPLE 4
Express the product in a + bi form: A 13 2 12i B (12 1 3i)
A 13 2 12i B (12 1 3i)
Solution
5 13 (12 1 3i) 2 12i(12 1 3i)
5 4 1 i 2 6i 2 32i2
5 4 2 5i 2 32 (21)
5 82 1 32 2 5i
5 11
2 2 5i
Calculator ENTER:
Solution
DISPLAY:
Answer
11
2
1 3 1 2 2nd
(
2nd
i
)
MATH
ENTER
)
i
(
12 3
ENTER
(1/3-1/2i)(12+3i
)Frac
11/2-5i
2 5i
Dividing Complex Numbers
3i
We can write (2 1 3i) 4 (1 1 2i) as 21 1
1 2i. However, this is not a complex num-
1 3i
ber of the form a 1 bi. Can we write 12 1
2i as an equivalent fraction with a
rational denominator? In Chapter 3, we found a way to write a fraction with an
irrational denominator as an equivalent fraction with a rational denominator.
We can use a similar procedure to write a fraction with a complex denominator
as an equivalent fraction with a real denominator.
3i
To write the quotient 21 1
1 2i with a denominator that is a real number,
multiply numerator and denominator of the fraction by the complex conjugate
of the denominator. The complex conjugate of 1 1 2i is 1 2 2i.
2 1 3i
1 1 2i
2i
2 2 4i 1 3i 2 6i2
i 2 6i2
? 11 2
522
2 2i 5
12 2 (2i) 2
1 2 4i2
5
5
5
The quotient (2 1 3i) 4 (1 1 2i) 5
a 1 bi form.
2 1 3i
1 1 2i
5
2 2 i 2 6(21)
1 2 4(21)
8 2 i
5
8
1
5 2 5i
8
1
5 2 5 i is a complex
number in
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Operations with Complex Numbers
215
EXAMPLE 5
Express the quotient in a 1 bi form: 51 1 10i
2
2 2 3i
How to Proceed
Solution
(1) Multiply numerator and
denominator by 6 to express
the denominator in terms
of integers:
5 1 10i
5 1 10i 6
60i
5
? 5 303 1
2 4i
1 2 2i
1 2 2i 6
2
3
2
3
(2) Multiply numerator and
denominator of the fraction
by the complex conjugate of
the denominator, 3 1 4i:
30 1 60i
3 2 4i
90 1 120i 1 180i 1 240i2
4i
? 33 1
2
2
1 4i 5
3 1 4
5
5
(3) Replace i2 by its equal, 21:
5
(4) Combine like terms:
90 1 120i 1 180i 1 240(21)
9 1 16
90 1 120i 1 180i 2 240
25
2150 1 300i
25
5 26 1 12i Answer
(5) Divide numerator and
denominator by 25:
Calculator ENTER:
Solution
DISPLAY:
(
2nd
5 10 2nd
i
)
i
)
(
1 2 2 3
ENTER
(5+10i)/(1/2-2/3
i)
-6+12i
Exercises
Writing About Mathematics
1. Tim said that the binomial x2 1 16 can be written as x2 2 16i2 and factored over the set of
complex numbers. Do you agree with Tim? Explain why or why not.
2. Joshua said that the product of a complex number and its conjugate is always a real number. Do you agree with Joshua? Explain why or why not.
Developing Skills
In 3–17, find each sum or difference of the complex numbers in a + bi form.
3. (6 1 7i) 1 (1 1 2i)
4. (3 2 5i) 1 (2 1 i)
5. (5 2 6i) 1 (4 1 2i)
6. (23 1 3i) 2 (1 1 5i)
7. (2 2 8i) 2 (2 1 8i)
8. (4 1 12i) 1 (24 2 2i)
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Quadratic Functions and Complex Numbers
9. (1 1 9i) 2 (1 1 2i)
12.
A 12
1
2i B
1
1
A 14
2
10. (10 2 12i) 2 (12 1 7i)
3
4i B
13.
15. A 232 1 53i B 2 A 94 2 15i B
A 23
2
1
6i B
1 (2 2 i)
i
16. A 14 1 12i B 1 A 7 1 10
B
11. (0 1 3i) 1 (0 2 3i)
14. (1 1 0i) 2 A 15 2 25i B
17.
A 37 1 13i B
1 A 16 2 78i B
In 18–25, write the complex conjugate of each number.
18. 3 1 4i
19. 2 2 5i
20. 28 1 i
21. 26 2 9i
22. 12 2 3i
23. 24 1 13i
24. 53 2 23i
25. p 1 2i
In 26–37, find each product.
26. (1 1 5i)(1 1 2i)
27. (3 1 2i)(3 1 3i)
28. (2 2 3i)2
29. (4 2 5i)(3 2 2i)
30. (7 1 i)(2 1 3i)
31. (22 2 i)(1 1 2i)
32. (24 2 i)(24 1 i)
33. (212 2 2i)(12 2 2i)
34. (5 2 3i)(5 1 3i)
35. (3 2
36.
37.
3
i) A 10
1
1
10i B
A 15
2
1
10i B (4
1 2i)
A 218 1 18i B (4 1 4i)
In 38–45, find the multiplicative inverse of each of the following in a + bi form.
38. 11 i
39. 21 4i
40. 21 1 2i
41. 3 2 3i
42. 12 2 14i
43. 2 2 12i
44. 56 1 3i
45. 9 1 pi
In 46–60, write each quotient in a + bi form.
46. (8 1 4i) 4 (1 1 i)
47. (2 1 4i) 4 (1 2 i)
48. (10 1 5i) 4 (1 1 2i)
49. (5 2 15i) 4 (3 2 i)
2i
50. 24 1
2 2i
2 5i
51. 10
2 1 6i
2i
52. 81 1
1 3i
i
53. 33 1
2 i
2i
54. 55 2
1 2i
1 3i
55. 12 3i
6i
56. 8 1
2i
2 i
57. 53 2 54i
58. 11 1 1i
59.
2 2 2i
3i
p 1 21i
1
1
1
1i
60. 7 5
i
Applying Skills
61. Impedance is the resistance to the flow of current in an electric circuit measured in ohms.
The impedance, Z, in a circuit is found by using the formula Z 5 V
I where V is the voltage
(measured in volts) and I is the current (measured in amperes). Find the impedance when
V 5 1.8 2 0.4i volts and I 5 20.3i amperes.
62. Find the current that will flow when V 5 1.6 2 0.3i volts and Z 5 1.5 1 8i ohms using the
formula Z 5 V
I.
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217
5-6 COMPLEX ROOTS OF A QUADRATIC EQUATION
When we used the quadratic formula to find the
roots of a quadratic equation, we found that some
quadratic equations have no real roots. For example, the quadratic equation x2 1 4x 1 5 5 0 has no
real roots since the graph of the corresponding
quadratic function f(x) 5 x2 1 4x 1 5, shown on
the right, has no x-intercepts. Can the roots of a
quadratic equation be complex numbers?
We will use the quadratic formula to find the
roots of the equation x2 1 4x 1 5 5 0. For this
equation, a 5 1, b 5 4, and c 5 5. The value of the
discriminant is
b2 2 4ac 5 42 2 4(1)(5)
y
f(x) x2 4x 5
1
x
O
1
or
24
Therefore, the roots are not real numbers.
2
24 6 "42 2 4(1)(5)
x 5 2b 6 "2ab 2 4ac 5
5 24 62 !24
2(1)
5 24 26 2i
5 22 6 i
The roots are the complex numbers 22 1 i and 22 2 i.
Therefore, every quadratic equation with rational coefficients has two complex roots. Those roots may be two real numbers. Complex roots that are real
may be rational or irrational and equal or unequal. Roots that are not real numbers are complex conjugates.
Note that complex roots that are not real are often referred to as imaginary
roots and complex roots of the form 0 1 bi 5 bi as pure imaginary roots. A quadratic equation with rational coefficients has roots that are pure imaginary only
if b 5 0 and b2 2 4ac , 0.
Let ax2 1 bx 1 c 5 0 be a quadratic equation and a, b, and c be rational
numbers with a 0:
The number
of x-intercepts
of the function is:
When the discriminant
b2 2 4ac is
The roots of the
equation are
. 0 and a perfect square
real, rational, and unequal
2
. 0 and not a perfect square
real, irrational, and unequal
2
50
real, rational, and equal
1
,0
imaginary numbers
0
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Quadratic Functions and Complex Numbers
EXAMPLE 1
Find the complex roots of the equation x2 1 12 5 6x.
Solution
How to Proceed
(1) Write the equation in standard form:
x2 1 12 5 6x
x2 2 6x 1 12 5 0
(2) Determine the values of a, b, and c:
(3) Substitute the values of a, b, and c
into the quadratic formula:
a 5 1, b 5 26, c 5 12
2
x 5 2b 6 "2ab 2 4ac
5
(4) Perform the computation:
(5) Write !212 in simplest form in
terms of i:
2(26) 6 " (26) 2 2 4(1)(12)
2(1)
2 48
5 6 6 !36
2
5 6 6 2!212
5 6 6 !42!3 !21
5 6 6 22i !3
5 3 6 i !3
Answer The roots are 3 1 i !3 and 3 2 i !3.
EXAMPLE 2
For what values of c does the equation 2x2 1 3x 1 c 5 0 have imaginary roots?
Solution
How to Proceed
(1) The equation has imaginary roots when
b2 2 4ac , 0. Substitute a 5 2, b 5 3:
(2) Isolate the term in c:
b2 2 4ac , 0
32 2 4(2)c , 0
9 2 8c , 0
28c , 29
(3) Solve the inequality. Dividing by a negative
number reverses the order of the inequality:
Answer The roots are imaginary when c . 98.
28c
28
. 29
28
c . 98
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219
Exercises
Writing About Mathematics
1. Emily said that when a and c are real numbers with the same sign and b 5 0, the roots of
the equation ax2 1 bx 1 c 5 0 are pure imaginary. Do you agree with Emily? Justify your
answer.
2. Noah said that if a, b, and c are rational numbers and b2 2 4ac , 0, then the roots of the
equation ax2 1 bx 1 c 5 0 are complex conjugates. Do you agree with Noah? Justify your
answer.
Developing Skills
In 3–14, use the quadratic formula to find the imaginary roots of each equation.
3. x2 2 4x 1 8 5 0
4. x2 1 6x 1 10 5 0
5. x2 2 4x 1 13 5 0
6. 2x2 1 2x 1 1 5 0
7. x2 1 10x 1 29 5 0
8. x2 1 8x 1 17 5 0
9. x2 2 2x 1 10 5 0
10. 4x2 2 4x 1 5 5 0
11. 4x2 1 4x 1 17 5 0
12. x2 1 5 5 4x
13. 4x – 7 5 x2
14. 2x = x2 1 3
5-7 SUM AND PRODUCT OF THE ROOTS OF
A QUADRATIC EQUATION
Writing a Quadratic Equation Given the
Roots of the Equation
We know that when a quadratic equation is in standard form, the polynomial
can often be factored in order to find the roots. For example, to find the roots of
2x2 2 5x 1 3 5 0, we can factor the polynomial.
2x2 2 5x 1 3 5 0
(2x 2 3)(x 2 1) 5 0
2x 2 3 5 0
x2150
2x 5 3
x51
x5
3
2
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Quadratic Functions and Complex Numbers
We can work backward on each of these steps to write a quadratic equation
with a given pair of factors. For example, write a quadratic equation whose roots
are 3 and 25:
(1) Let x equal each of the roots:
x53
(2) Write equivalent equations with
one side equal to 0:
x2350
(3) If equals are multiplied by equals
their products are equal:
x 5 25
x1550
(x 2 3)(x 1 5) 5 0(0)
x2 1 2x 2 15 5 0
We can check to show that 3 and 25 are the roots of x2 1 2x 2 15 5 0:
Check for x 5 3:
Check for x 5 25:
2
x2 1 2x 2 15 5 0
x 1 2x 2 15 5 0
?
(25)2 1 2(25) 2 15 5 0
?
?
25 2 10 2 15 5 0
(3)2 1 2(3) 2 15 5 0
?
9 1 6 2 15 5 0
050✔
050✔
We can use the same procedure for the general case. Write the quadratic
equation that has roots r1 and r2:
(1) Let x equal each of the roots:
(2) Write equivalent equations with
one side equal to 0:
(3) If equals are multiplied by equals,
their products are equal:
x 5 r1
x 5 r2
x 2 r1 5 0
x 2 r2 5 0
(x 2 r2)(x 2 r1) 5 0(0)
2
x 2 r1x 2 r2x 1 r1r2 5 0
x2 2 (r1 1 r2)x 1 r1r2 5 0
In the example given above, r1 5 3 and r2 5 25. The equation with these
roots is:
x2 2 (r1 1 r2)x 1 r1r2 5 0
x2 2 [3 1 (25)]x 1 3(25) 5 0
x2 1 2x 2 15 5 0
EXAMPLE 1
Write an equation with roots 1 13 !6 and 1 23 !6.
Solution r1 + r2 5 1
1 !6
3
1 1 23 !6 5 23
12 2 A !6B 2
6
5
512
5 25
r1r2 5 Q 1 13 "6 R Q 1 23 "6 R 5
9
9
9 5 29
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221
Therefore, the equation is x2 2 23x 2 59 5 0 or 9x2 2 6x 2 5 5 0.
x 5 Q 1 13 !6 R
Alternative (1) Let x equal each
of the roots:
Solution
x 5 Q 1 23 !6 R
x 2 Q 1 13 !6 R 5 0
(2) Write equivalent
equations with
one side equal
to 0:
cx
(3) If equals are
multiplied by
equals, their
products are
equal:
x 2 Q 1 23 !6 R 5 0
2 Q 1 13 !6 R d ? c x 2 Q 1 23 !6 R d 5 0
x2 2 Q 1 23 !6 Rx 2 Q 1 13 !6 Rx 1 Q 1 13 !6 RQ 1 23 !6 R 5 0
6
x2 2 23x 1 1 2
50
9
5
2
2
x 2 3x 2 9 5 0
(4) Simplify:
or
9x2 2 6x 2 5 5 0
Answer 9x2 2 6x 2 5 5 0
The Coefficients of a
Quadratic Equation and Its Roots
Compare the equation x2 2 (r1 1 r2)x 1 r1r2 5 0 derived previously with the
standard form of the quadratic equation, ax2 1 bx 1 c 5 0.
(1) Divide each side of the equation by a to write the general equation as an
equivalent equation with the coefficient of x2 equal to 1.
a 2
ax
1 bax 1 ac 5 a0
x2 1 bax 1 ac 5 0
(2) Compare the equation in step 1 with x2 2 (r1 1 r2)x 1 r1r2 5 0 by equating corresponding coefficients.
b
a
5 2 (r1 1 r2)
or
c
a
2ba 5 r1 1 r2
5 r1r2
We can conclude the following:
If r1 and r2 are the roots of ax2 1 bx 1 c 5 0, then:
2ba is equal to the sum of the roots r1 1 r2.
c
a
is equal to the product of the roots r1r2.
Thus, we can find the sum and product of the roots of a quadratic equation without actually solving the equation.
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Quadratic Functions and Complex Numbers
Note that an equivalent equation of any quadratic equation can be written
with the coefficient of x2 equal to 1. For example, 3x2 2 5x 1 2 5 0 and
x2 2 53x 1 23 5 0 are equivalent equations.
•
•
The sum of the roots of 3x2 2 5x 1 2 5 0 is 2 A 253 B or 53.
The product of the roots of 3x2 2 5x 1 2 5 0 is 23.
EXAMPLE 2
If one root of the equation x2 2 3x 1 c 5 0 is 5, what is the other root?
Solution The sum of the roots of the equation is 2ba 5 223
1 5 3.
Let r1 5 5. Then r1 1 r2 5 3 or r2 5 22.
Answer The other root is 22.
EXAMPLE 3
Find two numbers whose sum is 4 and whose product is 5.
Solution Let the numbers be the roots of the equation ax2 1 bx + c 5 0.
The sum of the roots is 2ba 5 4 and the product of the roots is ac 5 5.
We can choose any convenient value of a. Let a 5 1.
Then:
2b1 5 4 or b 5 24
c
1
5 5 or
c55
The equation is x2 2 4x 1 5 5 0.
Use the quadratic equation to find the roots.
2
x 5 2b 6 "2ab 2 4ac
x5
2(24) 6 " (24) 2 2 4(1)(5)
2(1)
5 4 6 2"24 5 4 62 2i 5 2 6 i
The numbers are (2 1 i) and (2 2 i).
Check The sum (2 1 i) 1 (2 2 i) 5 4. ✔
The product (2 1 i)(2 2 i) 5 22 2 i 2 5 4 2 (21) 5 4 1 1 5 5. ✔
Answer (2 1 i) and (2 2 i)
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223
Exercises
Writing About Mathematics
1. The roots of a quadratic equation with rational coefficients are p 6 !q. Write the equation
in standard form in terms of p and q.
2. Adrien said that if the roots of a quadratic equation are 21 and 43, the equation is
4x2 2 5x 1 32 5 0. Olivia said that the equation is 8x2 2 10x 1 3 5 0. Who is correct?
Justify your answer.
Developing Skills
In 3–17, without solving each equation, find the sum and product of the roots.
3. x2 1 x 1 1 5 0
4. x2 1 4x 1 5 5 0
5. 2x2 2 3x 2 2 5 0
6. 5x2 1 2x 2 10 5 0
7. 3x2 2 6x 1 4 5 0
8. –x2 1 3x 1 1 5 0
9. 8x 1 12 5 x2
12. x2 2 14 5 0
15. 8x2 2 9 5 26x
10. 4x2 5 2x 1 9
11. 2x2 2 8 5 5x
13. x2 1 1 5 0
14. x2 1 2x 5 0
16. x2 1 2x 2 3 5 0
2
17. 3x 1 33x 1 5 5 0
In 18–27, one of the roots is given. Find the other root.
18. x2 1 15x 1 c 5 0; 25
19. x2 2 8x 1 c 5 0; 23
20. 2x2 1 3x 1 c 5 0; 1
21. –4x2 2 5x 1 c 5 0; 2
22. –6x2 1 2x 1 c 5 0; 21
23. 6x2 2 x 1 c 5 0; 223
24. m2 2 4m 1 n 5 0; 3
25. z2 1 2z 1 k 5 0; 34
26. x2 1 bx 1 3 5 0; 1
27. –7w2 1 bw 2 5 5 0; 21
28. One root of the equation 23x2 1 9x 1 c 5 0 is !2.
a. Find the other root.
b. Find the value of c.
c. Explain why the roots of this equation are not conjugates.
29. One root of the equation 2x2 2 11x 1 c 5 0 is !3.
a. Find the other root.
b. Find the value of c.
c. Explain why the roots of this equation are not conjugates.
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Quadratic Functions and Complex Numbers
In 30–43, write a quadratic equation with integer coefficients for each pair of roots.
30. 2, 5
31. 4, 7
32. –3, 4
33. –2, 21
34. 23, 3
35. 21, 72
36. 34, 238
37. 1, 0
38. 2 1 !3, 2 2 !3
39. 1 12 !5, 1 22 !5
40. 21 13 !3, 21 23 !3
41. 3 1 i, 3 2 i
42. 3 22 2i, 3 12 2i
43. 32i, 232i
Applying Skills
44. One root of a quadratic equation is three more than the other. The sum of the roots is 15.
Write the equation.
45. The difference between the roots of a quadratic equation is 4i. The sum of the roots is 12.
Write the equation.
46. Write a quadratic equation for which the sum of the roots is equal to the product of the
roots.
47. Use the quadratic formula to prove that the sum of the roots of the equation
ax2 1 bx + c 5 0 is 2ba and the product is ac .
48. A root of x2 1 bx 1 c 5 0 is an integer and b and c are integers. Explain why the root must
be a factor of c.
5-8 SOLVING HIGHER DEGREE POLYNOMIAL EQUATIONS
The graph of the polynomial function
f(x) 5 x3 2 3x2 2 4x 1 12 is shown at the right. We
know that the real roots of a polynomial function
are the x-coordinates of the points at which the
graph intersects the x-axis, that is, the values of x for
which y 5 f(x) 5 0. A polynomial function of degree
three can have no more than three real roots. The
graph appears to intersect the x-axis at 22, 2, and 3,
but are these exact values of the roots? We know
that we can find the rational roots of a polynomial
function by factoring. We can also show that these
are exact roots by substituting the roots in the equation to show that they satisfy the equation.
y
2
O
2
x
f(x) x3 3x2 4x 12
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225
Solving Higher Degree Polynomial Equations
Solve by factoring:
Find a common binomial factor by
first factoring the first two terms and
then factoring the last two terms of
the polynomial. Then factor out the
common binomial factor.
f(x) 5 x3 2 3x2 2 4x 1 12
Solve by substitution:
x 5 3: f(x) 5 x3 2 3x2 2 4x 1 12
0 5 33 2 3(3)2 2 4(3) 1 12
0 5 27 2 27 2 12 1 12
050
x 5 2: f(x) 5 x3 2 3x2 2 4x 1 12
0 5 x2(x 2 3) 2 4(x – 3)
0 5 23 2 3(2)2 2 4(2) 1 12
0 5 (x 2 3)(x2 2 4)
0 5 8 2 12 2 8 1 12
Now factor the difference of two
squares.
0 5 (x 2 3)(x2 2 4)
050
x 5 22: f(x) 5 x3 2 3x2 2 4x 1 12
0 5 (22)3 2 3(22)2 2 4(22) 1 12
0 5 (x 2 3)(x 2 2)(x 1 2)
x2350
x2250
x53
x52
x1250
0 5 28 2 12 1 8 1 12
050
x 5 22
y
Not every polynomial function has rational
roots. The figure at the right shows the graph of
the polynomial function f(x) 5 x3 1 2x2 2 2x.
From the graph we see that this function appears
to have a root between 23 and 22, a root at 0, and
a root between 0 and 1. Can we verify that 0 is a
root and can we find the exact values of the other
two roots?
(1) Let f(x) 5 0. Factor the polynomial
if possible. We can factor out the
common factor x:
(2) The factor x2 1 2x 2 2 cannot be
factored in the set of integers. Set
each factor equal to 0. Use the
quadratic formula to solve the
quadratic equation:
(3) The roots of f(x) 5 x3 1 2x2 2 2x
are 0, 21 1 !3, and 21 2 !3:
1
O
1
x
f(x) x3 2x2 2x
f(x) 5 x3 1 2x2 2 2x
0 5 x3 1 2x2 2 2x
0 5 x(x2 1 2x 2 2)
x50
x2 1 2x 2 2 5 0
x5
5
5
22 6 "22 2 4(1)(22)
2(1)
22 6 !12
2
22 6 2 !3
2
5 21 6 !3
We can use a calculator to approximate the values of 21 1 !3 and
21 2 !3:
–1 2 !3 22.7 and 21 1 !3 0.7
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These are values between 23 and 22 and between 0 and 1 as seen on the graph
on the previous page. The function has three real roots. One of the roots, 0, is
rational and the other two roots, 21 1 !3 and 21 2 !3, are irrational roots.
Note that in factoring a polynomial of degree three or greater, we look for
the following factors:
1. a common monomial factor.
2. the binomial factors of a trinomial.
3. a common binomial factor.
4. factors of the difference of two squares.
EXAMPLE 1
Find three roots of the function f(x) 5 x3 2 2x2 1 9x 2 18.
Solution
How to Proceed
(1) Try to factor the polynomial.
There is no common monomial
factor. Look for a common
binomial factor:
(2) Let f(x) 5 0:
(3) Set each factor equal to 0. and
solve each equation for x:
x3 2 2x2 1 9x 2 18
5 x2(x 2 2) 1 9(x 2 2)
5 (x 2 2)(x2 1 9)
f(x) 5 x3 2 2x2 1 9x 2 18
0 5 (x 2 2)(x2 1 9)
x2250
x52
x2 1 9 5 0
x2 5 29
x 5 6 !29
x 5 6 !9 !21
x 5 63i
Answer The roots are 2, 3i, and 23i.
EXAMPLE 2
Is 2 is a root of the function f(x) 5 x4 2 5x2 1 x 1 2?
Solution If 2 is a root of f(x) 5 x4 2 5x2 1 x 1 2, then f(2) 5 0.
f(2) 5 24 2 5(2)2 1 2 1 2 5 16 2 20 1 2 1 2 5 0 ✔
Answer 2 is a root of f(x) 5 x4 2 5x2 1 x 1 2.
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227
Exercises
Writing About Mathematics
1. Sharon said that if f(x) is a polynomial function and f(a) 5 0, then a is a root of the function. Do you agree with Sharon? Explain why or why not.
2. Jordan said that if the roots of a polynomial function f(x) are r1, r2, and r3, then the roots of
g(x) 5 f(x 2 a) are r1 1 a, r2 1 a, and r3 1 a. Do you agree with Jordan? Explain why or
why not.
Developing Skills
In 3–18, find all roots of each given function by factoring or by using the quadratic formula.
3. f(x) 5 x3 1 7x2 1 10x
4. f(x) 5 2x3 1 2x2 2 4x
5. f(x) 5 x3 1 3x2 1 4x 1 12
6. f(x) 5 x3 2 x2 1 3x 2 3
7. f(x) 5 2x3 2 3x2 2 2x 1 3
8. f(x) 5 22x3 1 6x2 2 x 1 3
9. f(x) 5 x4 2 5x2 1 4
10. f(x) 5 x4 1 5x2 1 4
11. f(x) 5 x4 2 81
12. f(x) 5 16x4 2 1
13. f(x) 5 x4 2 10x2 1 9
14. f(x) 5 x5 2 x4 2 2x3
15. f(x) 5 x3 2 18x
16. f(x) 5 (2x2 1 x 2 1)(x2 2 3x 1 4)
17. f(x) 5 (x2 2 1)(3x2 1 2x 1 1)
18. f(x) 5 x3 1 2x2 2 x 2 2
In 19–28: a. Find f(a) for each given function. b. Is a a root of the function?
19. f(x) 5 x4 2 1 and a 5 1
20. f(x) 5 x3 1 4x, and a 5 22
21. f(x) 5 5x2 1 4x 1 1 and a 5 21
22. f(x) 5 2x3 1 x 2 24 and a 5 23
23. f(x) 5 x4 1 x2 1 x 1 1 and a 5 0
24. f(x) 5 2x3 1 3x2 2 1 and a 5 12
25. f(x) 5 x3 2 3x2 1 x 2 3 and a 5 i
27. f(x) 5 x3 2 2x 1 3 and a 5 2 1 i
26. f(x) 5 x4 2 2x2 1 x and a 5 !3
28. f(x) 5 25x3 1 5x2 1 2x 1 3 and a 5 32i
Applying Skills
29. a. Verify by multiplication that (x 2 1)(x2 1 x 1 1) 5 x3 2 1.
b. Use the factors of x3 2 1 to find the three roots of f(x) 5 x3 2 1.
3
c. If x3 2 1 5 0, then x3 5 1 and x 5 !
1. Use the answer to part b to write the three
cube roots of 1. Explain your reasoning.
d. Verify that each of the two imaginary roots of f(x) 5 x3 2 1 is a cube root of 1.
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30. a. Verify by multiplication that (x 1 1)(x2 2 x 1 1) 5 x3 1 1.
b. Use the factors of x3 1 1 to find the three roots of f(x) 5 x3 1 1.
3
c. If x3 1 1 5 0, then x3 5 21 and x 5 !
21. Use the answer to part b to write the three
cube roots of 21. Explain your reasoning.
d. Verify that each of the two imaginary roots of f(x) 5 x3 1 1 is a cube root of 21.
31. Let f(x) 5 x3 1 3x2 2 2x 2 6 and g(x) 5 2f(x) 5 2x3 1 6x 2 4x 212 be two cubic polynomial functions.
a. How does the graph of f(x) compare with the graph of g(x)?
b. How do the roots of f(x) compare with the roots of g(x)?
c. In general, if p(x) 5 aq(x) and a . 0, how does the graph of p(x) compare with the
graph of q(x)?
d. How do the roots of p(x) compare with the roots of q(x)?
Hands-On Activity
The following activity will allow you to evaluate a function for any constant and test possible roots
of a function. This process is called synthetic substitution. Let f(x) 5 x4 2 3x3 1 x2 2 2x 1 3.
Find f(2):
(1) List the coefficients of the terms of the function and
the constant to be tested as shown:
1 23
1
22
3
2
(2) Bring down the first coefficient as shown and then
multiply it by the constant 2 and write the product
under the second coefficient:
1 23
2
1
1
22
3
2
(3) Add the second coefficient and the product found
in step 2:
1 23
2
1 21
1
22
3
2
(4) Multiply this sum by the constant 2 and place the
product under the next coefficient:
1 23
1 22
2 22
1 21
3
2
(5) Add the coefficient and the product found in step 4:
1 23
1 22
2 22
1 21 21
3
2
(6) Repeat steps 4 and 5 until a final sum is found using
the last coefficient:
1 23
1 22
3 2
2 22 22 28
1 21 21 24 25
The final number is 25. Therefore, f(2) 5 25. Use a calculator to verify that this is true. Repeat
the steps for f(1). The final number should be 0. This means that f(1) 5 0 and that 1 is a root of the
function.
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229
Repeat the process for the following function, using the numbers 23, 22, 21, 1, 2, and 3 for each
function. Find the roots of the function.
a. f(x) 5 x3 2 2x2 2 x 1 2
b. f(x) 5 x3 23x2 2 4x 1 12
c. f(x) 5 x3 2 7x 2 6 5 x3 1 0x2 2 7x 2 6 (Include 0 in the list of coefficients.)
d. f(x) 5 x4 2 5x2 1 4 5 x4 1 0x3 2 5x2 1 0x 1 4
5-9 SOLUTIONS OF SYSTEMS OF EQUATIONS AND INEQUALITIES
A system of equations is a set of two or more equations. The system is consistent
if there exists at least one common solution in the set of real numbers. The solution set of a consistent system of equations can be found using a graphic method
or an algebraic method.
Solving Quadratic-Linear Systems
A system that consists of a quadratic function whose graph is a parabola and a
linear function whose graph is a straight line is a quadratic-linear system in two
variables. In the set of real numbers, a quadratic-linear system may have two
solutions, one solution, or no solutions. Each solution can be written as the coordinates of the points of intersection of the graph of the parabola and the graph
of the line.
y
y
y
O
x
Two Solutions
O
One Solution
x
O
x
No Solution
Recall that for the function y 5 ax2 1 bx + c, the x-coordinate of the turning point and the equation of the axis of symmetry is x 5 2b
2a . When a is positive,
the parabola opens upward and has a minimum value of y as shown in the figures above. When a is negative, the parabola opens downward and has a maximum value of y as shown in the following example.
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Quadratic Functions and Complex Numbers
We can use either a graphic method or an algebraic method to find the common solutions of y 5 2x2 1 6x 2 3 and y 2 x 5 1.
Graphic Method
(1) Graph the parabola on the calculator.
Choose a reasonable viewing window
and enter 2x2 1 6x 2 3 as Y1.
ENTER: Y
CLEAR
2 1 6 X,T,,n
(-)
X,T,,n
^
3 GRAPH
(2) On the same set of axes, draw the graph
of y 2 x 5 1. Write the equation in slopeintercept form: y 5 x 1 1 and enter it
as Y2.
ENTER: Y
CLEAR
X,T,,n
1 GRAPH
(3) The common solutions are the coordinates
of the points of intersection of the graphs.
We can find the intersection points by using
the intersect function. Press 2nd CALC
5 ENTER ENTER to select both curves.
When the calculator asks you for a guess,
move the cursor near one of the
intersection points using the arrow keys
and then press ENTER . Repeat this
process to find the other intersection
point.
*
Intersection
Y=2
X=1
*
Intersection
Y=5
X=4
The common solutions are (1, 2) and (4, 5).
Note: The graphic method only gives exact values when the solutions are rational numbers. If the solutions are irrational, then the calculator will give approximations.
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Solutions of Systems of Equations and Inequalities
Algebraic Method
(1) Solve the linear equation for one of the
variables in terms of the other. We will solve
for y in terms of x:
(2) Replace y in the quadratic equation by its
value in terms of x from the linear
equation:
(3) Write the equation in standard form:
(4) Factor the trinomial:
(5) Set each factor equal to 0 and solve for x:
(6) Use the simplest equation in x and y to find
the corresponding value of y for each value
of x:
y2x51
y5x11
y 5 2x2 1 6x 2 3
x 1 1 5 2x2 1 6x 2 3
x2 2 5x 1 4 5 0
(x 2 4)(x 2 1) 5 0
x2450 x2150
x54
x51
y=x11
y=x11
y5411 y5111
y55
y52
The common solutions are x 5 4, y 5 5 and x 5 1, y 5 2.
EXAMPLE 1
y
For the given system of equations, compare the solution
that can be read from the graph with the algebraic solution.
y 5 x2 2 4
y 5 22x
1
Solution From the graph, the solutions appear to be (23.2, 6.5) and
(1.2, 22.5).
O
1
x
We can find exact values by using an algebraic solution.
(1) Substitute the value of y from
the second equation in the first
equation:
(2) Write the equation in standard
form:
(3) The polynomial x2 1 2x 2 4
cannot be factored in the set
of integers. Use the quadratic
formula with a 5 1, b 5 2,
c 5 24:
(4) Use the linear equation to find
y for each value of x:
22x 5 x2 2 4
0 5 x2 1 2x – 4
2
x 5 2b 6 "2ab 2 4ac
5
5
22 6 "22 2 4(1)(24)
2(1)
22 6 !4 1 16
2
5 21 6 !5
y 5 22x
y 5 22A21 2 !5B
y 5 2 1 2!5
y 5 22x
y 5 22A21 1 !5B
y 5 2 2 2 !5
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Quadratic Functions and Complex Numbers
The exact solutions are A21 2 !5, 2 1 2!5B and A21 1 !5, 2 2 2!5B . The
rational approximations of these solutions, rounded to the nearest tenth, are
the solutions that we found graphically.
EXAMPLE 2
Use an algebraic method to find the solution set of the equations:
y 5 2x2 2 4x 2 5
3x 2 y 5 1
Solution (1) Solve the linear equation for y:
3x 2 y 5 1
2y 5 23x 1 1
y 5 3x 2 1
(2) Substitute for y in terms of x in
the quadratic equation:
(3) Write the equation in standard
form and solve for x:
y 5 2x2 2 4x 2 5
3x 2 1 5 2x2 2 4x 2 5
0 5 2x2 2 7x 2 4
0 5 (2x 1 1)(x 2 4)
2x 1 1 5 0
x2450
2x 5 21
x5
(4) Find the corresponding values
of y using the linear equation
solved for y:
x54
212
y 5 3x 2 1
y 5 3x 2 1
y 5 3 A 212 B 2 1
y5
y5
232
252
2
y 5 3(4) 2 1
2
2
y 5 12 2 1
y 5 11
Answer A 212, 252 B and (4, 11)
EXAMPLE 3
The graph shows the circle whose equation is
(x 2 2)2 1 (y 1 1)2 5 10 and the line whose
equation is y 5 x 2 1.
y
a. Read the common solutions from the
graph.
O
b. Check both solutions in both equations.
Solution a. The common solutions are x 5 21, y 5 22
and x 5 3, y 5 2.
1
x
1
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Solutions of Systems of Equations and Inequalities
233
b. Check x 5 21, y 5 22:
(x 2 2)2 1 (y 1 1)2 5 10
2 ?
2
(21 2 2) 1 (22 1 1) 5 10
?
(23)2 1 (21)2 5 10
y5x21
?
22 5 21 2 1
22 5 22 ✔
10 5 10 ✔
Check x 5 3, y 5 2:
(x 2 2)2 1 (y 1 1)2 5 10
2
25321
2 ?
252✔
(3 2 2) 1 (2 1 1) 5 10
2
y5x21
2 ?
(1) 1 (3) 5 10
10 5 10 ✔
Solving Quadratic Inequalities
You already know how to graph a linear inequality in two variables. The process
for graphing a quadratic inequality in two variables is similar.
EXAMPLE 3
Graph the inequality y 1 3 $ x2 1 4x.
Solution
How to Proceed
(1) Solve the inequality for y:
(2) Graph the corresponding quadratic function
y 5 x2 1 4x 2 3. Since the inequality is $ ,
use a solid curve to indicate that the parabola
is part of the solution:
(3) The values of y . x2 1 4x 2 3 are those in
the region above the parabola. Shade this region:
(4) Check: Use any convenient test point to verify
that the correct region has been shaded. Try (0, 0):
?
0 $ 02 1 4(0) 2 3
0 $ 23 ✔
The test point satisfies the equation. The shaded
region is the set of points whose coordinates
make the inequality true.
y 1 3 $ x2 1 4x
y $ x2 1 4x 2 3
y
1
2 O x
y
1
2 O x
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Quadratic Functions and Complex Numbers
In general:
The solution set of the inequality y + ax2 1 bx 1 c is the set of coordinates
of the points above the graph of y 5 ax2 1 bx 1 c, which is drawn as a dotted line to show that it is not part of the solution set.
The solution set of the inequality y # ax2 1 bx 1 c is the set of coordinates
of the points above the graph of y 5 ax2 1 bx 1 c, which is drawn as a solid
line to show that it is part of the solution set.
The solution set of the inequality y * ax2 1 bx 1 c is the set of coordinates
of the points below the graph of y 5 ax2 1 bx 1 c, which is drawn as a dotted line to show that it is not part of the solution set.
* ax2 1 bx 1 c is the set of coordinates
The solution set of the inequality y 2
of the points below the graph of y 5 ax2 1 bx 1 c, which is drawn as a solid
line to show that it is part of the solution set.
The following figures demonstrate some of these graphing rules:
O
y
y
y
x
y ax2 bx c
O
x
y ax2 bx c
O
y
x
y ax2 bx c
Graphs can also be used to find the solution
of a quadratic inequality in one variable. At the
right is the graph of the equation y 5 x2 1 4x 2 5.
The shaded region is the graph of y . x2 1 4x 2 5.
Let y 5 0. The graph of y 5 0 is the x-axis.
Therefore, the solutions of 0 . x2 1 4x 2 5 are the
x-coordinates of the points common to the graph
of y . x2 1 4x 2 5 and the x-axis. From the
graph on the right, we can see that the solution of
0 . x2 1 4x 2 5 or x2 1 4x 2 5 , 0 is the interval
25 , x , 1.
O
x
y ax2 bx c
y
1
1
O
x
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Solution of Systems of Equations and Inequalities
EXAMPLE 4
Solve the inequality 2x2 1 3x 1 4 # 0.
Solution
How to Proceed
(1) Let y 5 0. The inequality
2x2 1 3x 1 4 # 0 is equivalent to the
intersection of 2x2 1 3x 1 4 # y and
y 5 0. Graph the quadratic function
y 5 2x2 1 3x 1 4. Note that the graph
intersects the x-axis at 21 and 4. Shade
the area above the curve that is the graph
of y . 2x2 1 3x 1 4.
(2) The graph of y 5 0 is the x-axis. The
solution of 2x2 1 3x 1 4 # 0 is
the set of x-coordinates of the
points common to the graph of
y $ 2x2 2 3x 1 4 and the x-axis.
y
1
O
1
x
y
1
O
1
x
x # 21 or x $ 4
Answer x # 21 or x $ 4
Two Variable Inequalities and
the Graphing Calculator
The graphing calculator can also be used to
graph quadratic inequalities in two variables.
For instance, to graph the inequality of
Example 3, first rewrite the inequality with the
x variable on the right: y $ x2 1 4x 2 3. Enter
the quadratic function x2 1 4x 2 3 into Y1. Now
move the cursor over the \ to the left of Y1 using
the arrow keys. Since the inequality is $, press
ENTER until the cursor turns into . Press
GRAPH to graph the inequality.
Note: For $ and . inequalities, use
. For # and , inequalities, use
.
235
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Exercises
Writing About Mathematics
1. Explain the relationship between the solutions of y . ax2 1 bx + c and the solutions of
0 . ax2 + bx + c.
2. Explain why the equations y 5 x2 1 2 and y 5 22 have no common solution in the set of
real numbers.
Developing Skills
In 3–8, determine each common solution from the graph.
3.
4.
y
5.
y
y
2
1
O 1
O 2
x
1
x
O 1
6.
7.
y
8.
y
x
y
1
1
O1
x
O 1
1
O1
x
In 9–17, graph each system and determine the common solution from the graph.
9. y 5 x2 2 2x 2 1
y5x13
10. y = x2 1 2x
y 5 2x 1 1
11. 2x2 1 4x 2 y – 2 5 0
x1y54
12. y 5 2x2 1 6x 2 1
y5x13
13. y 5 2x2 1 2x 1 3
y2x53
14. y 5 2x2 2 6x 1 5
y5x12
15. x2 2 4x 2 y 1 4 5 0
1
16. x 2
5 x 16 12
y
y5x12
y
7
17. x 5 x 1
5
y 5 2x
y 5 4x 41 7
x
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237
In 18–35, find each common solution algebraically. Express irrational roots in simplest radical form.
18. y 5 x2 2 2x
y 5 3x
19. y 5 x2 1 4x
2x 2 y 5 1
20. y 5 x2 1 2x 1 3
x1y51
21. y 5 x2 2 8x 1 6
2x 2 y 5 10
22. 2x2 2 3x 1 3 2 y 5 0
y 2 2x 5 1
23. y 5 x2 2 4x 1 5
2y 5 x 1 6
24. y 5 2x2 2 6x 1 7
y5x14
25. x2 1 x 2 y 5 7
1
2x 5 y 1 2
26. y 5 4x2 2 6x 2 10
y 5 25 2 2x
27. y 5 x2 2 2x 1 1
35
y 5 29x 2
2
28. y 5 x2 1 4x 1 4
y 5 4x 1 6
29. 2x2 1 x 2 y 1 1 5 0
x2y1750
30. y 5 x2 1 2
2x 2 y 5 23
31. y 5 2x2 1 3x 1 4
y 5 11x 1 6
32. 5x2 1 3x 2 y 5 21
10x 1 1 5 y
33. x2 1 y2 5 24
x1y58
34. x2 1 y2 5 20
3x 2 y 5 10
35. x2 1 y2 5 16
y 5 2x
36. Write the equations of the graphs shown in Exercise 4 and solve the system algebraically.
(a 5 1)
37. Write the equations of the graphs shown in Exercise 6 and solve the system algebraically.
(a 5 21)
In 38–43, match the inequality with its graph. The graphs are labeled (1) to (6).
(1)
(2)
y
1
(4)
O
1
1
x
(5)
y
O
1
x
O
1
1
y
1
x
O
(6)
y
1
1
(3)
y
1
x
y
1
O
x
1O
x
38. x2 # 4 1 y
39. x2 . 22x 1 3 2 y
40. y 1 4x # x2
41. y . 22(x 2 2)2 1 3
42. y # 22(x 2 2)2 1 3
43. 4x2 2 2x 2 y 2 2 , 0
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Quadratic Functions and Complex Numbers
In 44–51: a. Graph the given inequality. b. Determine if the given point is in the solution set.
44. x2 # 2x 1 y; (5, 4)
45. x2 1 5 $ y; (21, 3)
46. x(x 2 13) . y; (25, 130)
47. x2 2 18 $ y 1 3x; (0, 218)
48. 2(x 2 3)2 1 3 , y; A 12, 6 B
49. –4(x 1 2)2 2 5 # y; A 1, 23 B
50. 4x2 2 4x , 3 1 y; A 0, 53 B
51. 6x2 1 x # 2 2 y; (10, 0)
Applying Skills
52. The area of a rectangular rug is 48 square feet. The length of the rug is 2 feet longer than
the width. What are the dimensions of the rug?
53. The difference in the lengths of the sides of two squares is 1 meter. The difference in the
areas of the squares is 13 square meters. What are the lengths of the sides of the squares?
54. The perimeter of a rectangle is 24 feet. The area of the rectangle is 32 square feet. Find the
dimensions of the rectangle.
55. The endpoints of a diameter of a circle are (0, 0) and (8, 4).
a. Write an equation of the circle and draw its graph.
b. On the same set of axes, draw the graph of x + y 5 4.
c. Find the common solutions of the circle and the line.
d. Check the solutions in both equations.
56. a. On the same set of axes, sketch the graphs of y 5 x2 2 4x 1 5 and y 5 2x 1 2.
b. Does the system of equations y 5 x2 2 4x 1 5 and y 5 2x 1 2 have a common solution in the set of real numbers? Justify your answer.
c. Find the solution set.
57. a. On the same set of axes, sketch the graphs of y 5 x2 1 5 and y 5 2x.
b. Does the system of equations y 5 x2 1 5 and y 5 2x have a common solution in the
set of real numbers? Justify your answer.
c. Does the system of equations y 5 x2 1 5 and y 5 2x have a common solution in the
set of complex numbers? If so, find the solution.
58. The graphs of the given equations have three points of intersection. Use an algebraic
method to find the three solutions of this system of equations:
y 5 x3 2 2x 1 1
y 5 2x 1 1
59. A soccer ball is kicked upward from ground level with an initial velocity of 52 feet per second. The equation h(t) 5 216t2 1 52t gives the ball’s height in feet after t seconds. To the
nearest tenth of a second, during what period of time was the height of the ball at least
20 feet?
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Chapter Summary
60. A square piece of cardboard measuring x inches by x inches is
to be used to form an open box by cutting off 2-inch squares,
as shown in the figure.
a. Express the length of the sides of the base of the box in
terms of x.
239
2 in.
2 in.
x
b. Write a function V(x) that represents the volume.
c. If the volume of the box must be greater than 128 cubic
inches, what are the minimum dimensions of the square
cardboard that can be used?
x
61. The profit function, in thousands of dollars, for a company that makes graphing calculators
is P(x) 5 25x2 1 5,400x 2 106,000 where x is the number of calculators sold in the millions.
a. Graph the profit function P(x).
b. How many calculators must the company sell in order to make a profit?
CHAPTER SUMMARY
The real roots of a polynomial function are the x-coordinates of the points
where the graph of the function intersects the x-axis.
Any quadratic equation with rational roots can be solved by factoring over
the set of integers.
Any quadratic equation can be solved by completing the square:
1. Write an equivalent equation with only the terms in x on one side.
2. Add the square of one-half the coefficient of x to both sides of the equation.
3. Take the square root of both sides of the equation.
4. Solve for x.
Any quadratic equation can be solved by using the quadratic formula:
2
x 5 2b 6 "2ab 2 4ac
When a, b, and c are rational numbers, the value of the discriminant,
b2 2 4ac, determines the nature of the roots and the number of x-intercepts of
the quadratic function.
The number
of x-intercepts
of the function is:
When the discriminant
b2 2 4ac is:
The roots of the
equation are:
. 0 and a perfect square
real, rational, and unequal
2
. 0 and not a perfect square
real, irrational, and unequal
2
50
real, rational, and equal
1
,0
imaginary numbers
0
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If r1 and r2 are the roots of a quadratic equation, the equation is
x2 2 (r1 1 r2)x 1 r1r2 5 0
For the quadratic equation ax2 1 bx + c 5 0:
• the sum of the roots is 2ba.
• the product of the roots is ac .
A number of the form a !21 5 ai is a pure imaginary number when a is a
non-zero real number.
For any integer n:
i 4n 5 1
i 4n11 5 i
i 4n12 5 21
i 4n13 5 2i
A complex number is a number of the form a + bi where a and b are real
numbers and i 5 !21. A complex number that is not a real number is an imaginary number.
The identity element for addition is the real number (0 1 0i) or 0.
The additive inverse of (a 1 bi) is (2a 2 bi).
The identity for multiplication is the real number (1 1 0i) or 1.
a
b
The multiplicative inverse of a 1 bi is a 11 bi or a2 1
b2 2 a2 1 b2 i.
The complex conjugate of a 1 bi is a 2 bi.
A polynomial of degree three or greater can be factored using:
• a common monomial factor.
• the binomial factors of a trinomial.
• a common binomial factor.
• factors of the difference of two squares.
A system of equations is a set of two or more equations. The system is
consistent if there exists at least one common solution in the set of real numbers. The solution set of a consistent system of equations can be found using a
graphic method or an algebraic method.
The graph of a quadratic function can be used to estimate the solution of a
quadratic inequality.
The solution set of the inequality y . ax2 1 bx 1 c is the set of coordinates
of the points above the graph of y 5 ax2 1 bx 1 c.
The solution set of the inequality y , ax2 1 bx 1 c is the set of coordinates
of the points below the graph of y 5 ax2 1 bx 1 c.
The solution set of 0 . ax2 1 bx 1 c is the set x-coordinates of points common to the graph of y . ax2 1 bx 1 c and the x-axis.
VOCABULARY
5-1 Completing the square
5-2 Quadratic formula
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5-3 Discriminant • Double root
5-4 i • Set of imaginary numbers • Pure imaginary number • Complex number
5-5 Complex conjugate
5-8 Synthetic substitution
5-9 Consistent • Quadratic-linear system
REVIEW EXERCISES
In 1–8, write each number in simplest form in terms of i.
1. !21
5. !24 1 !225
2. !216
6. !218 1 !232
3. !29
7. !264 Q #214 R
4. !212
8. !2128
!212
In 9–28, perform each indicated operation and express the result in a + bi form.
9. (2 1 3i) 1 (5 2 4i)
10. (1 1 2i) 1 (21 1 i)
11. (2 1 7i) 1 (2 2 7i)
12. (3 2 4i) 1 (23 1 4i)
13. (1 1 2i) 2 (5 1 4i)
14. (8 2 6i) 2 (22 2 2i)
15. (7 2 5i) 2 (7 1 5i)
16. (22 1 3i) 2 (22 2 3i)
17. (1 1 3i)(5 2 4i)
18. (2 1 6i)(3 2 i)
19. (9 2 i)(9 2 i)
20. 3i(4 2 2i)
2i
23. 2 1
2i
24. 21 11 2ii
21. A 12 2 i B (2 1 i)
3i
25. 22 1
2 3i
27. (1 1 4i)2
22. A 15 2 25i B (1 1 2i)
2 i
26. 31 2
i
28. (3 2 2i)2
In 29–36, find the roots of each equation by completing the square.
29. x2 2 7x 1 1 5 0
30. x2 2 x 2 12 5 0
31. x2 1 4x 1 5 5 0
32. x2 2 6x 2 10 5 0
33. x2 2 6x 1 10 5 0
x
5 2x 51 7
34. 12
36. 2x2 1 3x 2 2 5 0
35. 3x2 2 6x 1 6 5 0
In 37–44, find the roots of each equation using the quadratic formula. Express
irrational roots in simplest radical form.
37. x2 5 x 1 1
38. 2x2 2 2x 5 1
39. 5x2 5 2x 1 1
40. 4x2 2 12x 1 13 5 0
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41. x3 2 2x2 2 16x 1 32 5 0
42. x4 2 5x2 1 4 5 0
43. 0.1x2 1 2x 1 50 5 0
44. –3x2 1 12x 1 53 5 0
In 45–48, find the roots of the equation by any convenient method.
45. x3 2 6x2 1 4x 5 0
46. x4 2 10x2 1 9 5 0
47. 3x3 1 12x2 2 x 2 4 5 0
48. (3x 1 5)(x2 1 5x 2 6) 5 0
In 49–52, without graphing the parabola, describe the translation, reflection,
and/or scaling that must be applied to y 5 x2 to obtain the graph of each given
function.
49. y 5 x2 1 2x 1 2
50. y 5 x2 1 3x 1 10
51. y 5 4x2 2 6x 1 3
52. y 5 22x2 1 5x 2 10
53. The graph on the right is the parabola
y = ax2 1 bx + c with a, b, and c rational
numbers.
a. Describe the roots of the equation
0 5 ax2 1 bx + c if the discriminant is 49.
y
O
x
b. Describe the roots of the equation
0 5 ax2 1 bx + c if the discriminant is 32.
c. Can the discriminant be 0 or negative?
54. Let h(x) 5 220.5x2 1 300.1x 2 500. Use the discriminant to determine if
there exist real numbers, x, such that h(x) 5 325.
In 55–60, find each common solution graphically.
55. y 5 x2 2 4
x+y52
56. y 5 2x2 1 6x – 5
y5x21
57. y 5 2x2 2 8x
y 5 2x
58. x2 1 y2 5 25
x+y57
59. (x 2 3)2 1 y2 5 9
x1y56
60. (x 1 1)2 1 (y 2 2)2 5 4
y532x
In 61–70, find each common solution algebraically. Express irrational roots in
simplest radical form.
61. y 5 x2 2 2x 2 2
x+y54
62. y 5 x2 2 8
2x 2 y 5 0
63. x2 1 y 5 3x
x2y53
64. x9 5 y8
y 5 36 2 4x
65. x2 5 y 1 5x
2x 2 y 5 5
66. y 5 2x2 1 2
x5y21
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67. 3x2 2 2x 2 y 5 0
3x 1 5 5 2y
68. y 5 x2 2 3x 1 3
y 5 2x 1 1
69. y 5 x2 1 6x 1 4
y 5 2x 2 1
70. x2 1 y2 5 5
x2y2150
243
In 71–75, write each quadratic equation that has the given roots.
71. 23 and 5
72. 12 and 24
74. 5 1 3 !2 and 5 2 3 !2
73. !5 and 2!5
75. 6 1 2i and 6 2 2i
76. For what value of c does the equation x2 2 6x 1 c 5 0 have equal roots?
77. For what values of b does 2x2 1 bx 1 2 5 0 have imaginary roots?
78. For what values of c does x2 2 3x 1 c 5 0 have real roots?
In 79 and 80: a. Graph the given inequality. b. Determine if the given point is in
the solution set.
79. y 2 (x 1 2)2 1 5 $ 0; (2, 3)
80. 22x2 1 3x , y; A0, !5B
81. The perimeter of a rectangle is 40 meters and the area is 97 square meters.
Find the dimensions of the rectangle to the nearest tenth.
82. The manager of a theater is trying to determine the price to charge for
tickets to a show. If the price is too low, there will not be enough money to
cover expenses. If the price is too high, they may not get enough playgoers.
The manager estimates that the profit, P, in hundreds of dollars per show,
can be represented by P 5 2(t 2 12) 2 1 100 where t is the price of a
ticket in dollars. The manager is considering charging between $0 and $24.
a. Graph the profit function for the given range of prices.
b. The theater breaks even when profit is zero. For what ticket prices does
the theater break even?
c. What price results in maximum profit? What is the maximum profit?
83. Pam wants to make a scarf that is 20 inches longer than it is wide. She
wants the area of the scarf to be more than 300 square inches. Determine
the possible dimensions of the scarf.
Exploration
1. Find the three binomial factors of x3 2 x2 2 4x 1 4 by factoring.
2. One of these factors is (x 2 2). Write x3 2 x2 2 4x 1 4 as the product of
(x 2 2) and the trinomial that is the product of the other two factors.
3. Use synthetic substitution (see the Hands-On Activity on page 228) to
show that 2 is a root of f(x) 5 x3 2 x2 2 4x 1 4.
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4. Compare the coefficients of the trinomial from step 2 with the first three
numbers of the synthetic substitution in step 3.
In (1)–(4), test each of the given polynomial functions to see if the relationship
above appears to be true.
a. Use synthetic substitution to find a root r. (Try integers that are factors of
the constant term.)
b. Write (x 2 r) as one factor and use the first three numbers from the bottom line of the synthetic substitution as the coefficients of a trinomial
factor.
c. Multiply (x 2 r) times the trinomial factor written in step 2. Is the product
equal to the given polynomial?
d. Write the three factors of the given polynomial and the three roots of the
given function.
(1) f(x) 5 x3 2 6x2 1 11x 2 6
(3) f(x) 5 x3 2 5x2 1 7x 2 3
3
2
(2) f(x) 5 x 1 2x 2 5x 2 6
(4) f(x) 5 x3 2 4x2 1 5x 2 2
CUMULATIVE REVIEW
CHAPTERS 1–5
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. In simplest form, 3(1 2 2x)2 2 (x 1 2) is equal to
(1) 36x2 2 37x 1 7
(3) 12x2 2 13x 1 1
(2) 36x2 2 35x 1 11
(4) 12x2 2 13x 1 5
2. The solution set of x2 2 2x 2 8 5 0 is
(1) {2, 4}
(3) {2, 24}
(2) {22, 4}
(4) {22, 24}
3. In simplest form, the fraction
1
(1) 11
2
(2) 11
1 2 21
2 1 43
is equal to
3
(3) 11
4. The sum of A2 2 !18B and A24 1 !50B is
(1) 22 1 2 !2
(3) 22 1 4 !2
(2) 22 1 !32
(4) 22 2 8!2
6
(4) 11
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245
5. Which of the following products is a rational number?
(1) A2 1 !2B A2 1 !2B
(3) !2A2 1 !2B
(2) A2 1 !2B A2 2 !2B
(4) 2A2 2 !2B
6. If the graph of g(x) is the graph of f(x) 5 x2 moved 2 units to the right,
then g(x) is equal to
(1) x2 1 2
(2) x2 2 2
(3) (x 2 2)2
(4) (x 1 2)2
7. Which of the following is not a one-to-one function when the domain and
range are the largest set of real numbers for which y is defined?
(1) y 5 2x 1 3
(2) y 5 x2
(3) y 5 x
(4) y 5 !x
8. The equation of a circle with center at (1, 21) and radius 2 is
(1) (x – 1)2 1 (y 1 1)2 5 2
(3) (x 1 1)2 1 (y 2 1)2 5 2
(2) (x – 1)2 1 (y 1 1)2 5 4
(4) (x 1 1)2 1 (y 2 1)2 5 4
9. The equation of the axis of symmetry of the graph of y 5 2x2 2 4x 1 7 is
(1) x 5 1
(2) x 5 21
(3) x 5 2
(4) x 5 22
10. The graph of f(x) is the graph of f(x 2 4) under the translation
(1) T4,0
(2) T24,0
(3) T0,4
(4) T0,24
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
i
11. Write the quotient 23 2
2 i in a 1 bi form.
3
5 2x 31 1?
12. For what value(s) of x does x 2
2
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
!5
13. Write 33 1
with a rational denominator.
2 !5
14. What are the roots of the function f(x) 5 3 2 2x?
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Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. a. Sketch the graph of y 5 2x2 1 4x.
b. Shade the region that is the solution set of y # 2x2 1 4x.
16. Let f(x) 5 2x 1 4 and g(x) 5 x2.
a. Find f g(23).
b. Find an expression for h(x) 5 f g(x).
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CHAPTER
SEQUENCES
AND SERIES
When the Grant family purchased a computer for
$1,200 on an installment plan, they agreed to pay $100 each
month until the cost of the computer plus interest had been
paid. The interest each month was 1.5% of the unpaid balance. The amount that the Gant family still owed after each
payment is a function of the number of months that have
passed since they purchased the computer.
• At the end of month 1:
they owed (1,200 1 1,200 3 0.015 2 100) dollars or
(1,200 3 1.015 2 100) 5 $1,118.
• At the end of month 2:
they owed (1,118 1 1,118 3 0.015 2 100) dollars or
(1,118 3 1.015 2 100) 5 $1,034.77.
Each month, interest is added to the balance from the
previous month and a payment of $100 is subtracted.
We can express the monthly payments with the function
defined as {(n, f(n))}. Let the domain be the set of positive
integers that represent the number of months since the initial purchase.Then:
f(1) 5 1,118
f(2) 5 1,034.77
In general, for positive integers n:
f(n) 5 f(n 2 1) 3 1.015 2 100
This pattern continues until (f(n 2 1) 3 1.015) is
between 0 and 100, since the final payment would be
(f(n 2 1) 3 1.015) dollars.
In this chapter we will study sequential functions, such as
the one described here, whose domain is the set of positive
integers.
6
CHAPTER
TABLE OF CONTENTS
6-1 Sequences
6-2 Arithmetic Sequences
6-3 Sigma Notation
6-4 Arithmetic Series
6-5 Geometric Sequences
6-6 Geometric Series
6-7 Infinite Series
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
247
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6-1 SEQUENCES
A ball is dropped from height of 16 feet. Each time that it bounces, it reaches a
height that is half of its previous height. We can list the height to which the ball
bounces in order until it finally comes to rest.
After Bounce
Height (ft)
1
8
2
4
3
2
4
5
6
7
1
1
2
1
4
1
8
The numbers 8, 4, 2, 1, 12, 14, 18 form a sequence. A sequence is a set of numbers written in a given order. We can list these heights as ordered pairs of numbers in which each height is paired with the number that indicates its position in
the list. The set of ordered pairs would be:
{(1, 8), (2, 4), (3, 2), (4, 1), (5, 0.5), (6, 0.25), (7, 0.125)}
We associate each term of the sequence with the positive integer that specifies
its position in the ordered set. Therefore, a sequence is a special type of function.
DEFINITION
A finite sequence is a function whose domain is the set of integers
{1, 2, 3, . . . , n}.
y
The function that lists the height of the ball
after 7 bounces is shown on the graph at the right.
Often the sequence can continue without
end. In this case, the domain is the set of positive
integers.
1
O
1
x
DEFINITION
An infinite sequence is a function whose domain is the set of positive
integers.
The terms of a sequence are often designated as a1, a2, a3, a4, a5, . . . . If the
sequence is designated as the function f, then f(1) 5 a1, f(2) 5 a2, or in general:
f(n) 5 an
Most sequences are sets of numbers that are related by some pattern that
can be expressed as a formula. The formula that allows any term of a sequence,
except the first, to be computed from the previous term is called a recursive
definition.
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For example, the sequence that lists the heights to which a ball bounces
when dropped from a height of 16 feet is 8, 4, 2, 1, 0.5, 0.25, 0.125, . . . . In this
sequence, each term after the first is 12 the previous term. Therefore, for each
term after the first,
4 5 12 (8) , 2 5 12 (4) , 1 5 12 (2) , 0.5 5 12 (1) , 0.25 5 12 (0.5) , 0.125 5 12 (0.25) , . . . .
For n . 1, we can write the recursive definition:
an 5 12an 2 1
Alternatively, we can write the recursive definition as:
an 1 1 5 12an for n $ 1
A rule that designates any term of a sequence can often be determined from
the first few terms of the sequence.
EXAMPLE 1
a. List the next three terms of the sequence 2, 4, 8, 16, . . . .
b. Write a general expression for an.
c. Write a recursive definition for the sequence.
Solution a. It appears that each term of the sequence is a power of 2: 21, 22, 23, 24, . . . .
Therefore, the next three terms should be 25, 26, and 27 or 32, 64, and 128.
b. Each term is a power of 2 with the exponent equal to the number of the
term. Therefore, an 5 2n.
c. Each term is twice the previous term. Therefore, for n . 1, an 5 2an21.
Alternatively, for n $ 1, an11 5 2an.
Answers a. 32, 64, 128
b. an 5 2n
c. For n . 1, an 5 2an21 or for n $ 1, an11 5 2an.
EXAMPLE 2
Write the first three terms of the sequence an 5 3n 2 1.
Solution
a1 5 3(1) 2 1 5 2
a2 5 3(2) 2 1 5 5
Answer The first three terms of the sequence are 2, 5, and 8.
a3 5 3(3) 2 1 5 8
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Sequences and the Graphing Calculator
We can use the sequence function on the graphing calculator to view a sequence
for a specific range of terms. Evaluate a sequence in terms of variable from a
beginning term to an ending term. That is:
seq(sequence, variable, beginning term, ending term)
For example, we can examine the sequence an 5 n2 1 2 on the calculator.
To view the first 20 terms of the sequence:
ENTER:
2nd
X,T,⍜,n
DISPLAY:
LIST
,
1
,
5
X,T,⍜,n
20
)
x2
ⴙ
2
,
ENTER
s e q ( X 2+ 2 , X , 1 , 2 0 )
{3 6 11 18 27 3...
Note: We use the variable X instead of n to enter the sequence. We can also use
the left and right arrow keys to examine the terms of the sequence.
Exercises
Writing About Mathematics
1. Nichelle said that sequence of numbers in which each term equals half of the previous term
is a finite sequence. Randi said that it is an infinite sequence. Who is correct? Justify your
answer.
2. a. Jacob said that if an 5 3n 2 1, then an11 5 an 1 3. Do you agree with Jacob? Explain why
or why not.
b. Carlos said that if an 5 2n, then an11 5 2n11. Do you agree with Carlos? Explain why or
why not.
Developing Skills
In 3–18, write the first five terms of each sequence.
3. an 5 n
4. an 5 n 1 5
5. an 5 2n
6. an 5 n1
7. an 5 n2
8. an 5 20 2 n
9. an 5 3n
10. an 5 n2
11. an 5 2n 1 3
12. an 5 2n 2 1
15. an 5 2n
16. an 5 12 2 3n
n
13. an 5 n 1
1
4n
17. an 5 3
2
14. an 5 n 1
n
18. an 5 n2 1 i
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In 19–30: a. Write an algebraic expression that represents an for each sequence. b. Find the ninth
term of each sequence.
19. 2, 4, 6, 8, . . .
20. 3, 6, 9, 12, . . .
21. 1, 4, 7, 10, . . .
22. 3, 9, 27, 81, . . .
23. 12, 6, 3, 1.5, . . .
24. 7, 9, 11, 13, . . .
25. 10i, 8i, 6i, 4i, . . .
26. 12, 13, 14, 15, . . .
29. 1, 22, 3, 24, . . .
27. 12, 23, 34, 45, . . .
30. 1, !2, !3, 2, . . .
28. 2, 5, 10, 17, . . .
In 31–39, write the first five terms of each sequence.
31. a1 5 5, an = an21 1 1
32. a1 5 1, an11 5 3an
33. a1 5 1, an 5 2an21 1 1
34. a1 5 22, an 5 22an21
35. a1 5 20, an 5 an21 2 4
36. a1 5 4, an11 5 an 1 n
37. a2 5 36, an 5 13an21
38. a3 5 25, an11 5 2.5an
39. a5 5 12, an 5 a 1
n21
Applying Skills
40. Sean has started an exercise program. The first day he worked out for 30 minutes. Each day
for the next six days, he increased his time by 5 minutes.
a. Write the sequence for the number of minutes that Sean worked out for each of the
seven days.
b. Write a recursive definition for this sequence.
41. Sherri wants to increase her vocabulary. On Monday she learned the meanings of four new
words. Each other day that week, she increased the number of new words that she learned
by two.
a. Write the sequence for the number of new words that Sherri learned each day for a
week.
b. Write a recursive definition for this sequence.
42. Julie is trying to lose weight. She now weighs 180 pounds. Every week for eight weeks, she
was able to lose 2 pounds.
a. List Julie’s weight for each week.
b. Write a recursive definition for this sequence.
43. January 1, 2008, was a Tuesday.
a. List the dates for each Tuesday in January of that year.
b. Write a recursive definition for this sequence.
44. Hui started a new job with a weekly salary of $400. After one year, and for each year that
followed, his salary was increased by 10%. Hui left this job after six years.
a. List the weekly salary that Hui earned each year.
b. Write a recursive definition for this sequence.
45. One of the most famous sequences is the Fibonacci sequence. In this sequence,
a1 5 1, a2 5 1, and for n . 2, an = an22 1 an21. Write the first ten terms of this sequence.
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Hands-On Activity
The Tower of Hanoi is a famous problem that has challenged problem solvers throughout the ages.
The tower consists of three pegs. On one peg there are a number of disks of different sizes, stacked
according to size with the largest at the bottom. The task is to move the entire stack from one peg
to the other side using the following rules:
• Only one disk may be moved at a time.
• No disk may be placed on top of a
smaller disk.
Note that a move consists of taking the top
disk from one peg and placing it on another peg.
1. Use a stack of different-sized coins to model the Tower of Hanoi. What is the smallest number of moves needed if there are:
a. 2 disks?
b. 3 disks?
c. 4 disks?
d. n disks?
2. Write a recursive definition for the sequence described in Exercise 1, a–d.
6-2 ARITHMETIC SEQUENCES
The set of positive odd numbers, 1, 3, 5, 7, . . . , is a sequence. The first term, a1,
is 1 and each term is 2 greater than the preceding term. The difference between
consecutive terms is 2. We say that 2 is the common difference for the sequence.
The set of positive odd numbers is an example an arithmetic sequence.
DEFINITION
An arithmetic sequence is a sequence such that for all n, there is a constant d
such that an 2 an–1 5 d.
For an arithmetic sequence, the recursive formula is:
an 5 an21 1 d
An arithmetic sequence is formed when each term after the first is obtained
by adding the same constant to the previous term. For example, look at the first
five terms of the sequence of positive odd numbers.
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Arithmetic Sequences
a1 5 1
a1 5 1
a2 5 a1 1 2 5 1 1 2 5 3
a2 5 1 1 1(2)
a3 5 a2 1 2 5 3 1 2 5 5
a3 5 [1 1 1(2)] 1 2 5 1 1 2(2)
a4 5 a3 1 2 5 5 1 2 5 7
a4 5 [1 1 2(2)] 1 2 5 1 1 3(2)
a5 5 a4 1 2 5 7 1 2 5 9
a5 5 [1 1 3(2)] 1 2 5 1 1 4(2)
253
For this sequence, a1 5 1 and each term after the first is found by adding 2 to
the preceding term. Therefore, for each term, 2 has been added to the first term
one less time than the number of the term. For the second term, 2 has been added
once; for the third term, 2 has been added twice; for the fourth term, 2 has been
added three times. In general, for the nth term, 2 has been added n 2 1 times.
Shown below is a general arithmetic sequence with a1 as the first term and
d as the common difference:
a1, a1 1 d, a1 1 2d, a1 1 3d, a1 1 4d, a1 1 5d, . . . , a1 1 (n 2 1)d
If the first term of an arithmetic sequence is a1 and the common difference
is d, then for each term of the sequence, d has been added to a1 one less time
than the number of the term. Therefore, any term an of an arithmetic sequence
can be evaluated with the formula
an 5 a1 1 (n 2 1)d
where d is the common difference of the sequence.
EXAMPLE 1
For the arithmetic sequence 100, 97, 94, 91, . . . , find:
a. the common difference.
b. the 20th term of the sequence.
Solution a. The common difference is the difference between any term and the previous term:
97 2 100 5 23
or
94 2 97 5 23
or
91 2 95 5 23
The common difference is 23.
b. a20 5 100 1 (20 2 1)(23)
5 100 1 (19)(23)
5 100 2 57
5 43
The 20th term can also be found by writing the sequence:
100, 97, 94, 91, 88, 85, 82, 79, 76, 73, 70, 67, 64, 61, 58, 55, 52, 49, 46, 43
Answers a. d 5 23
b. a20 5 43
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EXAMPLE 2
Scott is saving to buy a guitar. In the first week, he put aside $42 that he received
for his birthday, and in each of the following weeks, he added $8 to his savings.
He needs $400 for the guitar that he wants. In which week will he have enough
money for the guitar?
Solution Let a1 5 42 and d 5 8.
The value for n for which an 5 400 is the week Scott will have enough money
to buy the guitar.
an 5 a1 1 (n 2 1)d
400 5 42 1 (n 2 1)(8)
400 5 42 1 8n 2 8
400 5 34 1 8n
366 5 8n
45.75 5 n
Scott adds money to his savings in $8 increments, so he will not have the
needed $400 in savings until the 46th week.
Answer 46th week
EXAMPLE 3
The 4th term of an arithmetic sequence is 80 and the 12th term is 32.
a. What is the common difference?
b. What is the first term of the sequence?
Solution a. How to Proceed
(1) Use an 5 a1 1 (n 2 1)d to
write two equation in two
variables:
80 5 a1 1 (4 2 1)d →
32 5 a1 1 (12 2 1)d → 2(32 5 a1 1 11d)
48 5 28d
(2) Subtract to eliminate a1:
26 5 d
(3) Solve for d:
b. Substitute in either equation to find a1.
80 5 a1 1 3(26)
80 5 a1 2 18
98 5 a1
Answers a. d 5 26
b. a1 5 98
80 5 a1 1 3d
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Arithmetic Means
We have defined the arithmetic mean of two numbers as their average, that is,
the sum of the numbers divided by 2. For example, the arithmetic mean of 4 and
28 is 4 12 28 5 32
2 5 16. The numbers 4, 16, and 28 form an arithmetic sequence
with a common difference of 12.
We can find three numbers between 4 and 28 that together with 4 and 28
form an arithmetic sequence: 4, a2, a3, a4, 28. This is a sequence of five terms.
1. Use the formula for a5 to find the common difference:
a5 5 a1 1 (n 2 1)d
28 5 4 1 (5 2 1)d
28 5 4 1 4d
24 5 4d
65d
2. Now use the recursive formula, an 5 an–1 1 d, and d 5 6, to write the
sequence:
4, 10, 16, 22, 28
The numbers 10, 16, and 22 are three arithmetic means between 4 and 28.
Looking at it another way, in any arithmetic sequence, any given term is the
average of the term before it and the term after it. Thus, an arithmetic mean is
any term of an arithmetic sequence.
EXAMPLE 4
Find five arithmetic means between 2 and 23.
Solution Five arithmetic means between 2 and 23 will form a sequence of seven terms.
Use the formula for a7 to find the common difference:
a7 5 a1 1 (n 2 1)d
23 5 2 1 (7 2 1)d
23 5 2 1 6d
21 5 6d
7
2
5d
Evaluate a2, a3, a4, a5, and a6:
a2 5 2 1 (2 2 1) 72
1
a2 5 11
2 5 52
a3 5 2 1 (3 2 1) 72
a3 5 9
a5 5 2 1 (5 2 1) 72
a5 5 16
a4 5 2 1 (4 2 1) 72
1
a4 5 25
2 5 122
a6 5 2 1 (6 2 1) 72
1
a6 5 39
2 5 192
Answer The five arithmetic means are 512, 9, 1212, 16, and 1912.
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Exercises
Writing About Mathematics
1. Virginia said that Example 3 could have been solved without using equations. Since there
are eight terms from a4 to a12, the difference between 80 and 32 has to be divided into eight
parts. Each part is 6. Since the sequence is decreasing, the common difference is 26. Then
using 26, work back from a4 to a1: 80, 86, 92, 98. Do you think that Virginia’s solution is better than the one given in Example 3? Explain why or why not.
2. Pedro said that to form a sequence of five terms that begins with 2 and ends with 12, you
should divide the difference between 12 and 2 by 5 to find the common difference. Do you
agree with Pedro? Explain why or why not.
Developing Skills
In 3–8, determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the
common difference.
3. 2, 5, 8, 11, 14, . . .
4. –3i, 21i, 1i, 3i, 5i, . . .
5. 1, 1, 2, 3, 5, 8, . . .
6. 20, 15, 10, 5, 0, . . .
7. 1, 2, 4, 8, 16, . . .
8. 1, 1.25, 1.5, 1.75, 2, . . .
In 9–14: a. Find the common difference of each arithmetic sequence. b. Write the nth term of each
sequence for the given value of n.
9. 3, 6, 9, 12, . . . , n 5 8
12.
3
1
2, 1, 2, 2, .
..,n57
10. 2, 7, 12, 17, . . . , n 5 12
11. 18, 16, 14, 12, . . . , n 5 10
13. 21, 23, 25, 27, . . . , n 5 10
14. 2.1, 2.2, 2.3, 2.4, . . . , n 5 20
15. Write the first six terms of the arithmetic sequence that has 12 for the first term and 42 for
the sixth term.
16. Write the first nine terms of the arithmetic sequence that has 100 as the fifth term and 80 as
the ninth term.
17. Find four arithmetic means between 3 and 18.
18. Find two arithmetic means between 1 and 5.
19. Write a recursive definition for an arithmetic sequence with a common difference of 23.
Applying Skills
20. On July 1, Mr. Taylor owed $6,000. On the 1st of each of the following months, he repaid
$400. List the amount owed by Mr. Taylor on the 2nd of each month starting with July 2.
Explain why the amount owed each month forms an arithmetic sequence.
21. Li is developing a fitness program that includes doing push-ups each day. On each day of
the first week he did 20 push-ups. Each subsequent week, he increased his daily push-ups by
5. During which week did he do 60 push-ups a day?
a. Use a formula to find the answer to the question.
b. Write the arithmetic sequence to answer the question.
c. Which method do you think is better? Explain you answer.
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257
22. a. Show that a linear function whose domain is the set of positive integers is an arithmetic
sequence.
b. For the linear function y 5 mx 1 b, y 5 an and x 5 n. Express a1 and d of the arithmetic sequence in terms of m and b.
23. Leslie noticed that the daily number of e-mail messages she received over the course of two
months form an arithmetic sequence. If she received 13 messages on day 3 and 64 messages
on day 20:
a. How many messages did Leslie receive on day 12?
b. How many messages will Leslie receive on day 50?
6-3 SIGMA NOTATION
Ken wants to get more exercise so he begins by walking for 20 minutes. Each
day for two weeks, he increases the length of time that he walks by 5 minutes.
At the end of two weeks, the length of time that he has walked each day is given
by the following arithmetic sequence:
20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85
The total length of time that Ken has walked in two weeks is the sum of the
terms of this arithmetic sequence:
20 1 25 1 30 1 35 1 40 1 45 1 50 1 55 1 60 1 65 1 70 1 75 1 80 1 85
This sum is called a series.
DEFINITION
A series is the indicated sum of the terms of a sequence.
The symbol S, which is the Greek letter sigma, is used to indicate a sum. The
number of minutes that Ken walked on the nth day is an 5 20 1 (n 2 1)(5). We
can write the sum of the number of minutes that Ken walked in sigma notation:
14
14
a an 5 a 20 1 (n 2 1)(5)
n51
n51
The “n 5 1” below S is the value of n for the first term of the series, and the
number above S is the value of n for the last term of the series. The symbol
14
a an can be read as “the sum of an for all integral values of n from 1 to 14.”
n51
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In expanded form:
14
a an 5 a1 1 a2 1 a3 1 a4 1 a5 1 a6 1 a7 1 a8 1 a9 1 a10 1 a11 1 a12 1 a13 1 a14
n51
14
a 20 1 (n 2 1)(5) 5 [20 1 0(5)] 1 [20 1 1(5)] 1 [20 1 2(5)] 1
c 1 [20 1 13(5)]
n51
5 20 1 25 1 30 1 ? ? ? 1 85
For example, we can indicate the sum of the first 50 positive even numbers
50
as a 2i.
i51
50
a 2i 5 2(1) 1 2(2) 1 2(3) 1 2(4) 1 ? ? ? 1 2(50)
i51
5 2 1 4 1 6 1 8 1 ? ? ? 1 100
Note that in this case, i was used to indicate the number of the term.
Although any variable can be used, n, i, and k are the variables most frequently
used. Be careful not to confuse the variable i with the imaginary number i.
50
The series a 2i is an example of a finite series since it is the sum of a finite
i51
number of terms. An infinite series is the sum of an infinite number of terms of
a sequence. We indicate that a series is infinite by using the symbol for infinity,
`. For example, we can indicate the sum of all of the positive even numbers as:
`
c 1 2i 1 c
a 2i 5 2 1 4 1 6 1
i51
EXAMPLE 1
7
Write the sum given by a (k 1 5) .
k51
7
Solution a (k 1 5) 5 (1 1 5) 1 (2 1 5) 1 (3 1 5) 1 (4 1 5) 1 (5 1 5) 1 (6 1 5) 1 (7 1 5)
k51
5 6 1 7 1 8 1 9 1 10 1 11 1 12 Answer
EXAMPLE 2
Write the sum of the first 25 positive odd numbers in sigma notation.
Solution The positive odd numbers are 1, 3, 5, 7, . . . .
The 1st positive odd number is 1 less than twice 1, the 2nd positive odd number is 1 less than twice 2, the 3rd positive odd number is 1 less than twice 3. In
general, the nth positive odd number is 1 less than twice n or an 5 2n 2 1.
25
The sum of the first 25 odd numbers is a (2n 2 1) . Answer
n51
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259
EXAMPLE 3
Use sigma notation to write the series 12 1 20 1 30 1 42 1 56 1 72 1 90 1 110
in two different ways:
a. Express each term as a sum of two numbers, one of which is a square.
b. Express each term as a product of two numbers.
Solution a. The terms of this series can be written as 32 1 3, 42 1 4, 52 1 5, . . ., 102 1 10,
or, in general, as n2 1 n with n from 3 to 10.
10
The series can be written as a (n2 1 n) .
n53
b. Write the series as
3(4) 1 4(5) 1 5(6) 1 6(7) 1 7(8) 1 8(9) 1 9(10) 1 10(11).
The series is the sum of n(n 1 1) from n 5 3 to n 5 10.
10
The series can be written as a n(n 1 1) .
n53
10
Answers a. a (n2 1 n)
n53
10
b. a n(n 1 1)
n53
EXAMPLE 4
Use sigma notation to write the sum of the reciprocals of the natural numbers.
Solution The reciprocals of the natural numbers are 1, 12, 13, 14, c, n1 .
Since there is no largest natural number, this sequence has no last term.
Therefore, the sum of the terms of this sequence is an infinite series. In sigma
notation, the sum of the reciprocals of the natural numbers is:
`
1
a n Answer
n51
Finite Series and the Graphing Calculator
The graphing calculator can be used to find the sum of a finite series. We use the
sum( function along with the seq( function of the previous section to evaluate
37
a series. For example, to evaluate a k(k 11 2) on the calculator:
k51
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STEP 1. Enter the sequence into the calculator and store it in list L1.
ENTER:
2nd
X,T,⍜,n
37
DISPLAY:
LIST
)
ⴙ
2
)
STO佡
1 ⴜ
5
2nd
(
)
,
L1
ENTER
(
X,T,⍜,n
1
,
X,T,⍜,n
,
seq(1/(X(X+2)),X
,1,37) >L1
{.3333333333 .1 ...
STEP 2. Use the sum( function to find the sum.
ENTER:
DISPLAY:
2nd
LIST
5
2nd
ENTER
L1
sum(L1
.7240215924
The sum is approximately equal to 0.72.
Exercises
Writing About Mathematics
8
1. Is the series given in Example 3 equal to a f (n 1 2) 2 1 (n 1 2)g ? Justify your answer.
n51
10
2. Explain why a k1 is undefined.
k50
Developing Skills
In 3–14: a. Write each arithmetic series as the sum of terms. b. Find each sum.
10
3. a 3n
5
4. a (2k 2 2)
n51
k51
6
10
6. a n3
7. a (100 2 5k)
n51
k51
5
10
9. a (n2 1 2i)
10. a (21) hh
n52
h51
10
5
12. a 2ki
k51
13. a (5 2 4k)
k53
4
5. a k2
k51
10
8. a (3n 2 3)
n55
15
11. a f4n 2 (n 1 1)g
n55
5
14. a (22n) n 1 1
n50
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261
In 15–26, write each series in sigma notation.
15. 3 1 5 1 7 1 9 1 11 1 13 1 15
16. 1 1 6 1 11 1 16 1 21 1 26 1 31 1 36
17. 11 1 22 1 33 1 44 1 55
18. 100 1 95 1 90 1 85 1 c 1 5
19. 3 1 6 1 9 1 12 1 15 1 c 1 30
1
1
1 32
20. 1 1 12 1 14 1 18 1 16
9
21. 21 1 23 1 34 1 45 1 56 1 67 1 78 1 89 1 10
1
1
1
1
1
1 2!
1 3!
1 4!
1 5!
22. 1!
3
5
4
1 81
2 243
23. 213 1 29 2 27
1
1
1
1
1
1
24. 1 3
2 1 2 3 3 1 3 3 4 1 4 3 5 1 5 3 6 1 6 3 7
25. 12 1 22 1 32 1 42 1 52 1 c
3
5
4
26. 13 1 29 1 27
1 81
1 243
1c
Applying Skills
n
n
27. Show that a kai 5 k a ai.
i51
n
i51
n
n
28. Show that a (ai 1 bi) 5 a ai 1 a bi.
i51
i51
i51
29. In a theater, there are 20 seats in the first row. Each row has 3 more seats than the row
ahead of it. There are 35 rows in the theater.
a. Express the number of seats in the nth row of the theater in terms of n.
b. Use sigma notation to represent the number of seats in the theater.
30. On Monday, Elaine spent 45 minutes doing homework. On the remaining four days of the
school week, she spent 15 minutes longer doing homework than she had the day before.
a. Express the number of minutes Elaine spent doing homework on the nth day of the
school week.
b. Use sigma notation to represent the total number of minutes Elaine spent doing
homework from Monday to Friday.
31. Use the graphing calculator to evaluate the following series to the nearest hundredth:
(1) a A 1 1 n1 B
50
n51
18
5
(2) a 1 1
k
k51
20
(3) a
n51
(n 2 1)(21) n
n
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6-4 ARITHMETIC SERIES
In the last section, we wrote the sequence of minutes that Ken walked each day
for two weeks:
20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85
Since the difference between each pair of consecutive times is a constant, 5,
the sequence is an arithmetic sequence. The total length of time that Ken
walked in two weeks is the sum of the terms of this sequence:
20 1 25 1 30 1 35 1 40 1 45 1 50 1 55 1 60 1 65 1 70 1 75 1 80 1 85
In general, if a1, a1 1 d, a1 1 2d, . . . , an 1 (n 2 1)d is an arithmetic sequence
with n terms, then:
n
c 1 fa 1 (n 2 1)dg
1
a fa1 1 (i 2 1)dg 5 a1 1 (a1 1 d) 1 (a1 1 2d) 1
i51
This sum is called an arithmetic series.
DEFINITION
An arithmetic series is the indicated sum of the terms of an arithmetic
sequence.
Once a given series is defined, we can refer to it simply as S (for sigma). Sn
is called the nth partial sum and represents the sum of the first n terms of the
sequence.
We can find the number of minutes that Ken walked in 14 days by adding
the 14 numbers or by observing the pattern of this series. Begin by writing the
sum first in the order given and then in reverse order.
S14 5 20 1 25 1 30 1 35 1 40 1 45 1 50 1 55 1 60 1 65 1 70 1 75 1 80 1 85
S14 5 85 1 80 1 75 1 70 1 65 1 60 1 55 1 50 1 45 1 40 1 35 1 30 1 25 1 20
Note that for this arithmetic series:
a1 1 a14 5 a2 1 a13 = a3 1 a12 = a4 1 a11 = a5 1 a10 = a6 1 a9 5 a7 1 a8
Add the sums together, combining corresponding terms. The sum of each
pair is 105 and there are 14
2 or 7 pairs.
S14 5 20 1 25 1 30 1 35 1 40 1 45 1 50 1 55 1 60 1 65 1 70 1 75 1 80 1 85
S14 5 85 1 80 1 75 1 70 1 65 1 60 1 55 1 50 1 45 1 40 1 35 1 30 1 25 1 20
2S14 5 1051105110511051105110511051105110511051105110511051105
2S14 5 14(105)
← Write the expression in factored form.
S14 5 7(105)
← Divide both sides by 2.
S14 5 735
← Simplify.
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Arithmetic Series
263
Therefore, the total number of minutes that Ken walked in 14 days is 7(105)
or 735 minutes.
Does a similar pattern exist for every arithmetic series? Consider the general arithmetic series with n terms, an 5 a1 1 (n 2 1)d. List the terms of the
series in ascending order from a1 to an and in descending order from an to a1.
Ascending Order
Descending Order
a1
an 5 a1 1 (n 2 1)d
a2 5 a1 1 d
an–1 5 a1 1 (n 2 2)d
[a1 1 d] 1 [a1 1 (n 2 2)d] 5 2a1 1 (n 2 1)d
a3 5 a1 1 2d
an–2 5 a1 1 (n 2 3)d
[a1 1 2d] 1 [a1 1 (n 2 3)d] 5 2a1 1 (n 2 1)d
(
Sum
a1 1 [a1 1 (n 2 1)d] 5 2a1 1 (n 2 1)d
(
an–2 5 a1 1 (n 2 3)d
a3 5 a1 1 2d
an–1 5 a1 1 (n 2 2)d
a2 5 a1 1 d
an 5 a1 1 (n 2 1)d
a1
(
[a1 1 (n 2 3)d] 1 [a1 1 2d] 5 2a1 1 (n 2 1)d
[a1 1 (n 2 2)d] 1 [a1 1 d] 5 2a1 1 (n 2 1)d
[a1 1 (n 2 1)d] 1 a1 5 2a1 1 (n 2 1)d
In general, for any arithmetic series with n terms there are n2 pairs of terms
whose sum is a1 1 an:
2Sn 5 n(a1 1 an) 5 n(2a1 1 (n 2 1)d)
Sn 5 n2 (a1 1 an) 5 n2 (2a1 1 (n 2 1)d)
EXAMPLE 1
a. Write the sum of the first 15 terms of the arithmetic series 1 1 4 1 7 1 c
in sigma notation.
b. Find the sum.
Solution a. For the related arithmetic sequence, 1, 4, 7, . . . , the common difference is 3.
Therefore,
an 5 1 1 (n 2 1)(3)
5 3n 2 2
15
The series is written as a 3n 2 2. Answer
n51
b. Use the formula Sn 5
n
2 (2a1
1 (n 2 1)d) with a1 5 1, n 5 15, and d 5 3.
S15 5 15
2 f2(1) 1 (15 2 1)(3)g
5 15
2 f2 1 14(3)g
5 15
2 f2 1 42g
5 15
2 f44g
5 330 Answer
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Note: Part b can also be solved by using the formula Sn 5 n2 (a1 1 an) . First find
a15.
S15 5 15
2 (1 1 43)
a15 5 a1 1 (n 2 1)d
5 1 1 (15 2 1)(3)
5 15
2 (44)
5 1 1 14(3)
5 330
5 1 1 42
5 43
Notice that for this series there are 712 pairs. The first 7 numbers are paired with
the last 7 numbers, and each pair has a sum of 44. The middle number, 22, is
paired with itself, making half of a pair.
EXAMPLE 2
The sum of the first and the last terms of an arithmetic sequence is 80 and the
sum of all the terms is 1,200. How many terms are in the sequence?
Solution
Sn 5 n2 (a1 1 an)
1,200 5 n2 (80)
1,200 5 40n
30 5 n
Answer There are 30 terms in the sequence.
Exercises
Writing About Mathematics
1. Is there more than one arithmetic series such that the sum of the first and the last terms is
80 and the sum of the terms is 1,200? Justify your answer.
2. Is 1 1 1 1 2 1 3 1 5 1 8 1 13 1 21 an arithmetic series? Justify your answer.
Developing Skills
In 3–8, find the sum of each series using the formula for the partial sum of an arithmetic series. Be
sure to show your work.
3. 2 1 4 1 6 1 8 1 10 1 12
4. 10 1 20 1 30 1 40 1 50 1 60
5. 3 1 1 2 1 2 3 2 5 2 7 2 9 2 11 2 13
6. 0i 1 4i 1 8i 1 12i 1 16i 1 20i
7. 0 1 13 1 23 1 1 1 43 1 53 1 2
8. !2 1 2 !2 1 3 !2 1 4 !2 1 c 1 15 !2
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265
In 9–18, use the given information to a. write the series in sigma notation, and b. find the sum of the
first n terms.
9. a1 5 3, an 5 39, d 5 4
10. a1 5 24, an 5 0, n 5 6
11. a1 5 24, an 5 0, d 5 26
12. a1 5 10, d 5 2, n 5 14
13. a1 5 2, d 5 21, n 5 15
14. a1 5 0, d 5 22, n 5 10
15. a1 5 31, d 5 13, n 5 12
16. a1 5 100, d 5 25, n 5 20
17. a10 5 50, d 5 2.5, n 5 10
18. a5 5 15, d 5 2, n 5 12
In 19–24: a. Write each arithmetic series as the sum of terms. b. Find the sum.
10
19. a 2k
k51
19
22. a (100 2 5i)
i50
8
20. a (3 1 k)
k52
25
23. a 22n
n51
9
21. a (20 2 2n)
n50
10
24. a (21 1 2n)
n51
Applying Skills
25. Madeline is writing a computer program for class. The first day she wrote 5 lines of code
and each day, as she becomes more skilled in writing code, she writes one more line than the
previous day. It takes Madeline 6 days to complete the program. How many lines of code
did she write?
26. Jose is learning to cross-country ski. He began by skiing 1 mile the first day and each day he
increased the distance skied by 0.2 mile until he reached his goal of 3 miles.
a. How many days did it take Jose to reach his goal?
b. How many miles did he ski from the time he began until the day he reached his goal?
27. Sarah wants to save for a special dress for the prom. The first month she saved $15 and each
of the next five months she increased the amount that she saved by $2. What is the total
amount Sarah saved over the six months?
28. In a theater, there are 20 seats in the first row. Each row has 3 more seats than the row
ahead of it. There are 35 rows in the theater. Find the total number of seats in the theater.
29. On Monday, Enid spent 45 minutes doing homework. On the remaining four days of the
school week she spent 15 minutes longer doing homework than she had the day before.
Find the total number of minutes Enid spent doing homework from Monday to Friday.
30. Keegan started a job that paid $20,000 a year. Each year after the first, he received a raise
of $600. What was the total amount that Keegan earned in six years?
31. A new health food store’s net income was a loss of $2,300 in its first month, but its net
income increased by $575 in each succeeding month for the next year. What is the store’s
net income for the year?
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Sequences and Series
6-5 GEOMETRIC SEQUENCES
Pete reuses paper that is blank on one side to write phone messages. One day
he took a stack of five sheets of paper and cut it into three parts and then cut
each part into three parts. The number of pieces of paper that he had after each
cut forms a sequence: 5, 15, 45. This sequence can be written as:
5, 5(3), 5(3)2
Each term of the sequence is formed by multiplying the previous term by 3,
or we could say that the ratio of each term to the previous term is a constant, 3.
If Pete had continued to cut the pieces of paper in thirds, the terms of the
sequence would be
5, 5(3), 5(3)2, 5(3)3, 5(3)4, 5(3)5, . . .
This sequence is called a geometric sequence.
DEFINITION
A geometric sequence is a sequence such that for all n, there is a constant r
a
such that a n 5 r. The constant r is called the common ratio.
n21
The recursive definition of a geometric sequence is:
an 5 an21r
When written in terms of a1 and r, the terms of a geometric sequence are:
a1, a2 = a1r, a3 = a1r2, a4 = a1r3, . . .
Each term after the first is obtained by multiplying the previous term by r. Therefore, each term is the
product of a1 times r raised to a power that is one less
than the number of the term, that is:
y
an 5 a1r n21
Since a sequence is a function, we can sketch the
function on the coordinate plane. The geometric
sequence 1, 1(2), 1(2)2, 1(2)3, 1(2)4 or 1, 2, 4, 8, 16 can
be written in function notation as {(1,1), (2, 2), (3, 4),
(4, 8), (5, 16)}. Note that since the domain is the set of
positive integers, the points on the graph are distinct
points that are not connected by a curve.
Many common problems can be characterized by
a geometric sequence. For example, if P dollars are
invested at a yearly rate of 4%, then the value of the
investment at the end of each year forms a geometric
sequence:
1
O
1
x
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Geometric Sequences
267
Year 1: P 1 0.04P 5 P(1 1 0.04) 5 P(1.04)
Year 2: P(1.04) 1 0.04(P(1.04)) 5 P(1.04)(1 1 0.04) 5 P(1.04)(1.04) 5 P(1.04)2
Year 3: P(1.04)2 1 0.04(P(1.04)2) 5 P(1.04)2(1 1 0.04) 5 P(1.04)2(1.04) 5 P(1.04)3
Year 4: P(1.04)3 1 0.04(P(1.04)3) 5 P(1.04)3(1 1 0.04) 5 P(1.04)3(1.04) 5 P(1.04)4
Year 5: P(1.04)4 1 0.04(P(1.04)4) 5 P(1.04)4(1 1 0.04) 5 P(1.04)4(1.04) 5 P(1.04)5
The terms P(1.04), P(1.04)2, P(1.04)3, P(1.04)4, . . . , P(1.04)n form a
geometric sequence in which a1 5 P(1.04) and r 5 1.04. The nth term is
an 5 a1r n–1 5 P(1.04)(1.04)n–1.
EXAMPLE 1
Is the sequence 4, 12, 36, 108, 324, . . . a geometric sequence?
324
36
108
Solution In the sequence, 12
4 5 3, 12 5 3, 36 5 3, 108 5 3, the ratio of any term to the
preceding term is a constant, 3, Therefore, 4, 12, 36, 108, 324, . . . is a
geometric sequence with a1 5 4 and r 5 3.
EXAMPLE 2
What is the 10th term of the sequence 4, 12, 36, 108, 324, . . . ?
Solution The sequence 4, 12, 36, 108, 324, . . . is a geometric sequence with a1 5 4 and
r 5 3. Therefore,
an 5 a1r n–1
a10 5 4(3)9
Use a calculator for the computation.
ENTER: 4
DISPLAY:
ⴛ
3 ^
9 ENTER
>
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4*3 9
78732
Answer 78,732
Geometric Means
20
4
In the proportion 20
, we say that 20 is the mean proportional between 4
5 100
and 100. These three numbers form a geometric sequence 4, 20, 100. The mean
proportional, 20, is also called the geometric mean between 4 and 100.
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Sequences and Series
Between two numbers, there can be any number of geometric means. For
example, to write three geometric means between 4 and 100, we want to form a
geometric sequence 4, a2, a3, a4, 100. In this sequence, a1 5 4 and a5 5 100.
an 5 a1rn 2 1
a5 5 a1r4
100 5 4r4
25 5 r4
4
4 4
!
25 5 !
r
r 5 6 !5
There are two possible values of r, 2!5 and 1!5. Therefore, there are two
possible sequences and two possible sets of geometric means.
One sequence is 4, 4 !5, 20, 20 !5, 100 with 4 !5, 20, and 20 !5 three geometric means between 4 and 100.
The other sequence is 4, 24 !5, 20, 220 !5, 100 with 24 !5, 20, and 220 !5
three geometric means between 4 and 100.
Note that for each sequence, the ratio of each term to the preceding term is
a constant.
4 !5
4
20
4 !5
20 !5
20
100
20 !5
24 !5
4
5 !5
5
5 !5
5 5 !5
5 5 !5
20
24 !5
220 !5
20
5 !5
5
5 !5
5 5 !5
5 5 !5
100
220 !5
5 2!5
!5
5
5 2!5
5 525
5 2!5
5 2!5
!5
5
5 2!5
5 525
5 2!5
EXAMPLE 3
Find four geometric means between 5 and 1,215.
Solution We want to find the missing terms in the sequence 5, a2, a3, a4, a5, 1,215. Use
the formula to determine the common ratio r for a1 5 5 and a6 5 1,215.
an 5 a1rn 2 1
a6 5 a1r5
1,215 5 5r5
243 5 r5
5
5
!243 5 !r5
r53
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Geometric Sequences
269
The four geometric means are 5(3) 5 15, 15(3) 5 45, 45(3) 5 135, and
135(3) 5 405.
Answer 15, 45, 135, and 405
Exercises
Writing About Mathematics
1. Autumn said that the answer to Example 2 could have been found by entering 4 ⴛ 3
ENTER on a calculator and then entering ⴛ 3 ENTER eight times to display
the sequence to the 9 terms after the first. Do you think that this is an easier way to
find the 10th term? Explain your answer.
2. Sierra said that 8, 8 !2, 16, 16 !2, 32 is a geometric sequence with three geometric means,
8 !2, 16, and 16 !2. Do you agree with Sierra? Justify your answer.
Developing Skills
In 3–14, determine whether each given sequence is geometric. If it is geometric, find r. If it is not
geometric, explain why it is not.
3. 4, 8, 16, 32, 64, . . .
4. 1, 5, 25, 125, 625, . . .
5. 3, 6, 9, 12, . . .
6. 21, 2, 8, 32, . . .
7. 1, 23, 9, 227, 81, . . .
8. 36, 12, 4, 34, . . .
10. 25, 2, 23, 1, 12, . . .
1
9. 1, 13, 19, 27
, ...
11. 1, 210, 100, 21,000, 10,000, . . .
12. 1, 0.1, 0.01, 0.001, 0.0001, . . .
13. 0.05, 20.1, 0.2, 20.4, . . .
14. a, a2, a3, a4, . . .
In 15–26, write the first five terms of each geometric sequence.
15. a1 5 1, r 5 6
16. a1 5 40, r 5 12
17. a1 5 2, r 5 3
18. a1 5 14, r 5 22
19. a1 5 1, r 5 !2
21. a1 5 21, a2 5 4
22. a1 5 100, a3 5 1
23. a1 5 1, a3 5 16
24. a1 5 1, a3 5 2
25. a1 5 1, a4 5 28
26. a1 5 81, a5 5 1
20. a1 5 10, a2 5 30
27. What is the 10th term of the geometric sequence 0.25, 0.5, 1, . . .?
28. What is the 9th term of the geometric sequence 125, 25, 5, . . .?
29. In a geometric sequence, a1 5 1 and a5 5 16. Find a9.
30. The first term of a geometric sequence is 1 and the 4th term is 27. What is the 8th term?
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Sequences and Series
31. In a geometric sequence, a1 5 2 and a3 5 16. Find a6.
32. In a geometric sequence, a3 5 1 and a7 5 9. Find a1.
33. Find two geometric means between 6 and 93.75.
34. Find three geometric means between 3 and 913
27.
35. Find three geometric means between 8 and 2,592.
Applying Skills
36. If $1,000 was invested at 6% annual interest at the beginning of 2001, list the geometric
sequence that is the value of the investment at the beginning of each year from 2001 to
2010.
37. Al invested $3,000 in a certificate of deposit that pays 5% interest per year. What is the
value of the investment at the end of each of the first four years?
38. In a small town, a census is taken at the beginning of each year. The census showed that
there were 5,000 people living in the town at the beginning of 2001 and that the population
decreased by 2% each year for the next seven years. List the geometric sequence that gives
the population of the town from 2001 to 2008. (A decrease of 2% means that the population
changed each year by a factor of 0.98.) Write your answer to the nearest integer.
39. It is estimated that the deer population in a park was increasing by 10% each year. If there
were 50 deer in the park at the end of the first year in which a study was made, what is the
estimated deer population for each of the next five years? Write your answer to the nearest
integer.
40. A car that cost $20,000 depreciated by 20% each year. Find the value of the car at the end
of each of the first four years. (A depreciation of 20% means that the value of the car each
year was 0.80 times the value the previous year.)
41. A manufacturing company purchases a machine for $50,000. Each year the company estimates the depreciation to be 15%. What will be the estimated value of the machine after
each of the first six years?
6-6 GEOMETRIC SERIES
DEFINITION
A geometric series is the indicated sum of the terms of a geometric sequence.
For example, 3, 12, 48, 192, 768, 3,072 is a geometric sequence with r 5 4. The
indicated sum of this sequence, 3 1 12 1 48 1 192 1 768 1 3,072, is a geometric series.
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Geometric Series
271
In general, if a1, a1r, a1r2, a1r3, . . . , a1rn21 is a geometric sequence with n
terms, then
n
a air
i21
5 a1 1 a1r 1 a1r 2 1 a1r 3 1 c 1 a1r n21
i51
is a geometric series.
Let the sum of these six terms be S6. Now multiply S6 by the negative of the
common ratio, 24. This will result in a series in which each term but the last is
the opposite of a term in S6.
S6 5 3 1 12 1 48 1 192 1 3,072
2 12 2 48 2 192 2 3,072 2 12,288
24SS6 5
23S6 5 3 2 12,288
S6 5
3 2 12,288
23
← Add the sums.
← Divide by 23.
5 21 1 4,096 5 4,095 ← Simplify.
Thus, S6 5 4,095. The pattern of a geometric series allows us to find a formula
for the sum of the series. To Sn, add 2rSn:
Sn 5 a1 1 a1r 1 a1r2 1 a1r3 1 c 1 a1rn21
2rSn 5 2 a1r 2 a1r2 2 a1r3 2 c 2 a1rn21 2 a1rn
Sn 2 rSn 5 a1 2 a1rn
Sn(1 2 r) 5 a1(1 2 rn)
Sn 5
a1 (1 2 r n)
12r
EXAMPLE 1
Find the sum of the first 10 terms of the geometric series 2 1 1 1 12 1 c.
Solution For the series 2 1 1 1 12 1 c, a1 5 2 and r 5 12 5
Sn 5
S10 5
Answer S10 5
1,023
256
1
2 5 1.
2
1
a (1 2 rn)
1
1 2 r
1,023
10
1
2 A 1 2 1,024
2 A 1,024 B
B
2 A 1 2 A 12 B B
1,023
1,023
5
5
5 2 A 1,024 B 3 12 5 256
1
1
1 2 21
2
2
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Sequences and Series
EXAMPLE 2
Find the sum of five terms of the geometric series whose first term is 2 and
whose fifth term is 162.
Solution Use a5 5 a1r521 to find r.
162 5 2r4
81 5 r4
!81 5 "r4
4
Sn 5
a1 (1 2 rn)
1 2 r
S5 5
2(1 2 35)
1 2 3
S5 5
2(1 2 (23) 5)
1 2 (23)
4
63 5 r
5
2(1 2 243)
22
5
5
2(2242)
22
2(1 1 243)
4
5
5 242
2(244)
4
5 122
Answer S5 5 242 or S5 5 122
Exercises
Writing About Mathematics
1. Casey said that the formula for the sum of a geometric series could be written as
a 2 ar
Sn 5 11 2 rn . Do you agree with Casey? Justify your answer.
2. Sherri said that Example 1 could have been solved by simply adding the ten terms of the
series on a calculator. Do you think that this would have been a simpler way of finding the
sum? Explain why or why not.
Developing Skills
In 3–14, find the sum of n terms of each geometric series.
3. a1 5 1, r 5 2, n 5 12
4. a1 5 4, r 5 3, n 5 11
5. a2 5 6, r 5 4, n 5 15
6. a1 5 10, r 5 10, n 5 6
7. a3 5 0.4, r 5 2, n 5 12
9. 5 1 10 1 20 1 c 1 an, n 5 8
11. 12 1 14 1 18 1 c 1 an, n 5 6
13. a1 5 4, a5 5 324, n 5 9
8. a1 5 1, r 5 13, n 5 10
10. 1 1 5 1 25 1 c 1 an, n 5 10
12. 2 2 8 1 32 2 c1 an, n 5 7
14. a1 5 1, a8 5 128, n 5 10
In 15–22: a. Write each sum as a series. b. Find the sum of each series.
16. 1 1 a A 13 B
5
5
15. 3 1 a 3(2) n
n51
17. 10 1 a 10 A 12 B
5
n51
n
n51
n
8
18. 26 1 a 26(4) n
n51
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Infinite Series
20. 1 1 a A 23 B
5
5
19. 1 1 a (22) n
n51
273
n
n51
21. 100 1 a 100 A 12 B
6
22. 281 1 a 281 A 213 B
6
k
k51
k
k51
23. Find the sum of the first six terms of the series 1 1 !3 1 3 1 3 !3 1 .
24. Find the sum of the first eight terms of a series whose first term is 1 and whose eighth term
is 625.
Applying Skills
25. A group of students are participating in a math contest. Students receive 1 point for their
first correct answer, 2 points for their second correct answer, 4 points for their third correct
answer, and so forth. What is the score of a student who answers 10 questions correctly?
26. Heidi deposited $400 at the beginning of each year for six years in an account that paid 5%
interest. At the end of the sixth year, her first deposit had earned interest for six years and
was worth 400(1.05)6 dollars, her second deposit had earned interest for five years and was
worth 400(1.05)5 dollars, her third deposit had earned interest for four years and was worth
400(1.05)4 dollars. This pattern continues.
a. What is the value of Heidi’s sixth deposit at the end of the sixth year? Express your
answer as a product and as a dollar value.
b. Do the values of these deposits after six years form a geometric sequence? Justify your
answer.
c. What is the total value of Heidi’s six deposits at the end of the sixth year?
27. A ball is thrown upward so that it reaches a height of 9 feet and then falls to the ground.
When it hits the ground, it bounces to 13 of its previous height. If the ball continues in this
way, bouncing each time to 13 of its previous height until it comes to rest when it hits the
ground for the fifth time, find the total distance the ball has traveled, starting from its highest point.
28. If you start a job for which you are paid $1 the first day, $2 the second day, $4 the third day,
and so on, how many days will it take you to become a millionaire?
6-7 INFINITE SERIES
Recall that an infinite series is a series that continues without end. That is, for a
series Sn 5 a1 1 a2 1 c 1 an, we say that n approaches infinity. We can also
`
write a an. However, while there may be an infinite number of terms, a series
n51
can behave in only one of three ways. The series may increase without limit,
decrease without limit, or approach a limit.
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Sequences and Series
CASE 1
The series increases without limit.
Consider an arithmetic series Sn 5 n2 (2a1 1 (n 2 1)d) .
If a1 5 1 and d 5 12:
S
18 n
n
1
16
Sn 5 2 S 2(1) 1 (n 2 1) A 2 B T
14
5 n2 S 42 1 n2 2 12 T
12
10
3n 1 n2
n
5 n2 S 3 1
5
8
T
2
4
6
4
n2
As the value of n approaches infinity, 3n 1
4
2
n
increases without limit. This series has no limit.
1 O1 2 3 4 5 6 7 8
We can see this by graphing the function
2
n2
Sn 5 3n 1
for
positive
integer
values.
4
CASE 2
The series decreases without limit.
For an arithmetic series Sn 5 n2 (2a1 1 (n 2 1)d) , if a1 5 1 and d 5 21:
Sn 5 n2 f2(1) 1 (n 2 1)(21)g
5
n
2 f2
2 n 1 1g 5
n
2 f3
2 ng
For this series, as the value of n approaches
n2
infinity, n2 f3 2 ng or 3n 2
decreases without
2
limit. This series also has no limit. We can see
n2
this by graphing the function Sn 5 3n 2
for
2
positive integer values.
CASE 3
2
Sn
n
O
1 1 2 3 4 5 6 7 8
2
24
26
28
210
212
214
216
218
The series approaches a limit.
a 2 a rn
For a geometric series Sn 5 11 2 r1 , if a1 5 1 and r 5 12:
Sn 5
n
1 2 1 A 12 B
1 2 21
5
Sn
n
1 2 A 12 B
1
2
2
n
n21
5 2 2 2 A 12 B 5 2 2 A 12 B
As n approaches infinity, A 12 B
0. Therefore, Sn 5 2 2
A 12 B
n21
n21
approaches
approaches
Sn 5 2 2 0 5 2 as n approaches infinity.
1.5
1
0.5
1
O
1 2 3 4 5 6 7
n
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Infinite Series
275
This series has a limit. The diagram on the bottom of page 274 shows how the
sum of n terms approaches 2 as the number of terms, n, increases.
In general,
An infinite arithmetic series has no limit.
An infinite geometric series has no limit when |r| + 1.
An infinite geometric series has a finite limit when |r| * 1.
When r , 1, rn approaches 0. Therefore:
Sn 5
a1 2 a1r n
12r
a
approaches 1 21 r as n approaches infinity.
EXAMPLE 1
Find a A 13 B .
`
n
n51
1
1 n
1 2
1 3
c
a A3B 5 3 1 A3B 1 A3B 1
`
Solution
n51
This is an infinite geometric series with a1 5 13 and r 5 13.
a 2 a rn
Using the formula 11 2 r1 ,
`
a
n51
n
n
A 13 B
2 A B
5 3 3 13
1 1
1
n
123
As n approaches infinity, A 13 B approaches 0. Therefore,
3
1 n
1
1
3
a A 3 B 5 1 2 1 5 3 3 2 5 2 Answer
n51
3
`
1
Series Containing Factorials
Series that are neither arithmetic nor geometric must be considered individually. Often the limit of a series can be found to be finite by comparing the terms
of the series to the terms of another series with a known limit. One such series
n
1
is a k!
.
k51
Recall factorials from previous courses. We write n factorial as follows:
DEFINITION
n! 5 n(n 2 1)(n 2 2)(n 2 3) c(3)(2)(1)
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Sequences and Series
n
1
1
1
1
1
Consider the series a k!
5 1!
1 2!
1 3!
1 c 1 n!
. This series can be
k51
shown to have a limit by comparing its terms to the terms of the geometric
series with a1 5 1 and r 5 12. Each term, an (n . 1), of this geometric series is
1
.
2n 2 1
n
Term of
Geometric Series
1
1
1
21
1
22
1
23
1
24
2
3
4
5
Term of Series
to Be Tested
1
1!
5 21
Comparison
of Terms
51
151
1
2!
1
1
523
1 52
1
1
1
3! 5 3 3 2 3 1 5 6
5 14
1
2
1
4
1
8
1
16
1
4!
5 18
1
1
54333
2 3 1 5 24
1
1
1
5! 5 5 3 4 3 3 3 2 3 1 5 120
1
5 16
5 12
. 16
1
. 24
1
. 120
1
1
1
5 n(n 2 1)(n 2
We can conclude that n!
2) c (2)(1) # 2n 2 1. Therefore:
1
1 1 1 1 1 c 1 1 , 1 1 1 1 12 1 13 1 c 1 n12 1
1!
2!
3!
n!
2
2
2
2
Previously, we found that for the geometric series with a1 5 1 and r 5 12, the
series approaches
1
1 2 21
`
1
or 2 as n approaches infinity. Therefore, a n!
, 2. Thus,
n51
this series is bounded above by 2.
1
1
1
1
To find a lower bound, notice that the series 1!
is the
1 2!
1 3!
1 c 1 n!
sum of positive terms. Therefore, for n . 3,
1
1
1
1
1
, 1!
1 2!
1 2!
1 3!
1 c 1 n!
3
1
1
1
c1 1
or
2 , 1! 1 2! 1 3! 1
n!
1
1!
`
1
Putting it all together, what we have shown is that for the series a n!
:
n51
3
2
1
1
1
1
* 1!
*2
1 2!
1 3!
1 c 1 n!
or
3
2
`
1
* a n!
*2
n51
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Infinite Series
277
The Number e
Let us add 1 to both sides of the inequality derived in the previous section. We
find that as n approaches infinity,
1
1
1
1
1 1 32 , 1 1 1!
1 2!
1 3!
1 c 1 n!
,112
5
1
1
1
1
c1 ,3
or
2 , 1 1 1! 1 2! 1 3! 1
n!
1
1
1
1
1 2!
1 3!
1 c 1 n!
The infinite series 1 1 1!
is equal to an irrational
number that is greater than 52 and less than 3. We call this number e.
Eighteenth-century mathematicians computed this number to many decimal places and assigned to it the symbol e just as earlier mathematicians computed to many decimal places the ratio of the length of the circumference of a
circle to the length of the diameter and assigned the symbol p to this ratio.
Therefore, we say:
`
1
1
1
1
1
c1
c5e
a n! 5 1 1 1! 1 2! 1 3! 1
(n 2 1)! 1
n50
This number has an important role in many different branches of mathematics. A calculator will give the value of e to nine decimal places.
ENTER:
ex
2nd
>
DISPLAY:
1 ENTER
e (1
2.718281828
EXAMPLE 2
Find, to the nearest hundredth, the value of 1 1 e2.
Solution Evaluate the expression on a calculator.
ENTER: 1
DISPLAY:
ⴙ
2nd
ex
2 ENTER
>
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1+e (2
8.389056099
Answer To the nearest hundredth, 1 1 e2 8.39.
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Exercises
Writing About Mathematics
1. Show that if the first term of an infinite geometric series is 1 and the common ratio is 1c, then
c
the sum is c 2
1.
2. Cody said that since the calculator gives the value of e as 2.718281828, the value of e can be
written as 2.71828, a repeating decimal and therefore a rational number. Do you agree with
Cody? Explain why or why not.
Developing Skills
In 3–10: a. Write each series in sigma notation. b. Determine whether each sum increases without
limit, decreases without limit, or approaches a finite limit. If the series has a finite limit, find that
limit.
3. 1 1 31 1 19 1 ? ? ?
1
4. 2 1 21 1 18 1 32
1???
5. 2 1 4 1 6 1 8 1 ? ? ?
1
6. 5 1 1 1 15 1 25
1???
7. 5 1 1 2 3 2 7 2 ? ? ?
8. 6 1 3 1 32 1 34 1 ? ? ?
1
1
1 3!
1 c 1 (n 11 1)! 1 c
9. 2!
10. 1 1 3 1 6 1 10 1 ? ? ?
In 11–16, an infinitely repeating decimal is an infinite geometric series. Find the rational number
represented by each of the following infinitely repeating decimals.
11. 1.11111 . . . 5 1 1 0.1 1 0.01 1 0.001 1 ? ? ?
12. 0.33333 . . . 5 0.3 1 0.03 1 0.003 1 0.0003 1 ? ? ?
13. 0.444444 . . .
14. 0.121212 . . . 5 0.12 1 0.0012 1 0.000012 1 ? ? ?
15. 0.242424 . . .
16. 0.126126126 . . .
17. The sum of the infinite series 1 1 12 1 12 1 13 1 c 1 21n is 2. Find values of n such that
2
2
226 2 1
.
2 2 an ,
25
2
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Chapter Summary
279
CHAPTER SUMMARY
A sequence is a set of numbers written in a given order. Each term of a
sequence is associated with the positive integer that specifies its position in the
ordered set. A finite sequence is a function whose domain is the set of integers
{1, 2, 3, . . . , n}. An infinite sequence is a function whose domain is the set of positive integers. The terms of a sequence are often designated as a1, a2, a3, . . . . The
formula that allows any term of a sequence except the first to be computed from
the previous term is called a recursive definition.
An arithmetic sequence is a sequence such that for all n, there is a constant
d such that an+1 2 an 5 d. For an arithmetic sequence:
an 5 a1 1 (n 2 1)d 5 an21 1 d
A geometric sequence is a sequence such that for all n, there is a constant r
a
such that na1 1 5 r. For a geometric sequence:
n
an 5 a1rn21 5 an21r
A series is the indicated sum of the terms of a sequence. The Greek letter S
is used to indicate a sum defined for a set of consecutive integer.
If Sn represents the nth partial sum, the sum of the first n terms of a
sequence, then
n
Sn 5 a ak 5 a1 1 a2 1 a3 1 ? ? ? 1 an
k51
For an arithmetic series:
Sn 5 n2 (a1 1 an) 5 n2 (2a1 1 (n 2 1)d)
For a geometric series:
Sn 5
a1 (1 2 r n)
12r
For a geometric series, if r , 1 and n approaches infinity:
a
Sn 5 1 21 r
`
or
a a1 r
n21
a
5 1 21 r
n51
As n approaches infinity:
`
1
1
1
1
1
c1
a n! 5 1 1 1! 1 2! 1 3! 1
(n 2 1)! 5 e
n50
The number e is an irrational number.
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Sequences and Series
VOCABULARY
6-1 Sequence • Finite sequence • Infinite sequence • Recursive definition
6-2 Common difference • Arithmetic sequence • Arithmetic means
6-3 Series • S • Sigma notation • Finite series • Infinite series • `
6-4 Arithmetic series • Sn • nth partial sum
6-5 Geometric sequence • Common ratio • Geometric mean
6-6 Geometric series
6-7 n factorial • e
REVIEW EXERCISES
In 1–6: a. Write a recursive formula for an. b. Is the sequence arithmetic, geometric, or neither? c. If the sequence is arithmetic or geometric, write an explicit
formula for an. d. Find a10.
1. 1, 5, 9, 13, . . .
2. 3, 1, 13, 19, . . .
3. 12, 11, 10, 9, . . .
4. 1, 3, 6, 10, 15, . . .
5. 1i, 3i, 7i, 15i, 31i, . . .
6. 2, 26, 18, 254, 162, . . .
In 7–12, write each series in sigma notation.
7. 1 1 3 1 6 1 10 1 15 1 21 1 28 1 36
8. 2 1 5 1 8 1 11 1 14 1 17 1 20
9. 4 1 6 1 8 1 10 1 12 1 14
10. 12 1 32 1 52 1 72 1 92 1 112
11. 1 2 2 1 3 2 4 1 5 2 6 1 7
12. 12 1 12 1 13 1 14 1 c
2
2
2
13. In an arithmetic sequence, a1 5 6 and d 5 5. Write the first five terms.
14. In an arithmetic sequence, a1 5 6 and d 5 5. Find a30.
15. In an arithmetic sequence, a1 5 5 and a4 5 23. Find a12.
16. In an arithmetic sequence, a3 5 0 and a10 5 70. Find a1.
17. In a geometric sequence, a1 5 2 and a2 5 5. Write the first five terms.
18. In a geometric sequence, a1 5 2 and a2 5 5. Find a10.
19. In a geometric sequence, a1 5 2 and a4 5 128. Find a6.
20. Write a recursive formula for the sequence 12, 20, 30, 42, 56, 70, 90, 110, . . . .
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Review Exercises
281
21. Write the first five terms of a sequence if a1 5 1 and an 5 4an–1 1 3.
22. Find six arithmetic means between 1 and 36.
23. Find three geometric means between 1 and 625.
24. In an arithmetic sequence, a1 5 1, d 5 8. If an 5 89, find n.
25. In a geometric sequence, a1 5 1 and a5 5 14. Find r.
4
26. Write a 3n as the sum of terms and find the sum.
n51
6
27. Write a (12 2 3k) as the sum of terms and find the sum.
k50
28. To the nearest hundredth, find the value of 3e3.
29. a. Write, in sigma notation, the series 3 1 1.5 1 0.75 1 0.375 1 0.1875 1 ? ? ?.
b. Determine whether the series given in part a increases without limit,
decreases without limit, or approaches a finite limit. If the series has a
finite limit, find that limit.
30. A grocer makes a display of canned tomatoes that are on sale. There are 24
cans in the first (bottom) layer and, after the first, each layer contains three
fewer cans than in the layer below. There are three cans in the top layer.
a. How many layers of cans are in the display?
b. If the grocer sold all of the cans in the display, how many cans of tomatoes did he sell?
31. A retail store pays cashiers $20,000 a year for the first year of employment
and increases the salary by $500 each year.
a. What is the salary of a cashier in the tenth year of employment?
b. What is the total amount that a cashier earns in his or her first 10 years?
32. Ben started a job that paid $40,000 a year. Each year after the first, his
salary was increased by 4%.
a. What was Ben’s salary in his eighth year of employment?
b. What is the total amount that Ben earned in eight years?
33. The number of handshakes that are exchanged if every person in a room
shakes hands once with every other person can be written as a sequence.
Let an21 be the number of handshakes exchanged when there are n 2 1
persons in the room. If one more persons enters the room so that there
are n persons in the room, that person shakes hands with n 2 1 persons,
creating n 2 1 additional handshakes.
a. Write a recursive formula for an.
b. If a1 5 0, write the first 10 terms of the sequence.
c. Write a formula for an in terms of n.
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Sequences and Series
Exploration
The graphing calculator can be used to graph recursive sequences and examine
the convergence, divergence, or oscillation of the sequence. A sequence converges
if as n goes to infinity, the sequence approaches a limit. A sequence diverges if
as n goes to infinity, the sequence increases or decreases without limit. A
sequence oscillates if as n goes to infinity, the sequence neither converges nor
diverges.
For example, to verify that the geometric sequence an 5 an21 A 12 B with
a1 5 5 converges to 0:
STEP 1. Set the calculator to sequence graphing
mode. Press MODE
and select Seq,
the last option in the fourth row.
Normal S c i E n g
Float 0 1 2 3 4 5 6 7 8 9
Radian D e g r e e
F u n c P a r P o l Seq
Connected D o t
Sequential S i m u l
Real a + b i r e u i
Full H o r i z G - T
>
14411C06.pgs
STEP 2. Enter the expression an21 A 12 B as
u(n 2 1) A 12 B into u(n). The function
name u is found above the 7 key.
ENTER:
Yⴝ
ⴚ
1
2nd
u
)
ⴛ
(
X,T,⍜,n
1 ⴜ 2.
Plot1 Plot 2 Plot3
n Mi n = 1
. . .u ( n ) = u ( n - 1 ) 1 /2
*
u ( n Mi n ) = { 5 }
. . .v ( n ) =
v ( n Mi n ) =
. . .w ( n ) =
STEP 3. To set the initial value of a1 5 5, set
u(nMin) 5 5.
ENTER:
5
(To set more than one initial value,
for example a1 5 5 and a2 5 6, set
u(nMin) 5 {5, 6}.)
STEP 4. Graph the sequence up to n 5 25.
Press WINDOW
nMax 5 25.
25 to set
–>
WINDOW
n Min=1
n Max=25
PlotStart=1
PlotStep=1
Xmin = - 10
Xmax=10
Xscl=1
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Cumulative Review
STEP 5. Finally, press
ZOOM
0
to graph the
sequence. You can use TRACE to
explore the values of the sequence. We
can see that the sequence approaches the
value of 0 very quickly.
283
.
.
.
.
.
.
.
.
...
............................
In 1–4: a. Graph each sequence up to the first 75 terms. b. Use the graph to make
a conjecture as to whether the sequence converges, diverges, or oscillates.
1. a1 5 1, a2 5 1, an = an–2 1 an–1 (Fibonacci sequence)
1
2. a1 5 5, an 5 an21
3. a1 5 100, an 5 12an21 1 100
4. a1 5 10, an 5 0.95an21 1 500(0.75) n
CUMULATIVE REVIEW
CHAPTERS 1–6
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. If the domain is the set of integers, then the solution set of
22 # x 1 3 , 7 is
(1) {22, 21, 0, 1, 2, 3, 4, 5, 6, 7}
(2) {22, 21, 0, 1, 2, 3, 4, 5, 6}
(3) {25, 24, 23, 22, 21, 0, 1, 2, 3, 4}
(4) {25, 24, 23, 22, 21, 0, 1, 2, 3}
2. The expression (a 1 3)2 2 (a 1 3) is equal to
(1) a2 2 a 1 6
(3) a2 1 5a 1 12
2
(2) a 1 5a 1 6
(4) a 1 3
2
2
2x2
3. In simplest form, x3 1 x 2
4
8 is equal to
(1) 11x242 6
(3) 11x241 6
2 18
(2) 11x 24
(4)
x(x 2 2) 2
24
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Sequences and Series
4. The sum of A3 2 !5B 1 A2 2 !45B is
(1) 5 2 4 !5
(2) 5 2 3 !5
5. The fraction 1 21 !2 is equal to
(1) 1 1 !2
(2) 21 1 !2
(3) 5 2 5 !2
(4) !2
(3) 21 2 !2
(4) 1 2 !2
6. The quadratic equation 3x2 2 7x 5 3 has roots that are
(1) real, rational, and equal.
(2) real, rational, and unequal.
(3) real irrational and unequal.
(4) imaginary.
7. A function that is one-to-one is
(1) y 5 2x 1 1
(2) y 5 x2 1 1
(3) y 5 x4 1 x
(4) y 5 x
8. The function y = x2 is translated 2 units up and 3 units to the left. The
equation of the new function is
(1) y 5 (x 2 3)2 1 2
(3) y 5 (x 1 3)2 1 2
(2) y 5 (x 2 3)2 2 2
(4) y 5 (x 1 3)2 2 2
9. Which of the following is a real number?
(1) i
(2) i2
(3) i3
(4) 2i
10. The sum of the roots of a quadratic equation is 24 and the product of the
roots is 5. The equation could be
(1) x2 2 4x 1 5 5 0
(3) x2 2 4x 2 5 5 0
2
(2) x 1 4x 2 5 5 0
(4) x2 1 4x 1 5 5 0
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicated the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work will receive only 1 credit.
11. What number must be added to each side of the equation x2 1 3x 5 0 in
order to make the left member the perfect square of a binomial?
12. Solve for x and check: 7 2 !x 1 2 5 4.
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Cumulative Review
285
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicated the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work will receive only 1 credit.
13. Express the fraction
(2 1 i) 2
i
in a 1 bi form.
14. Graph the solution set of the inequality 2x2 2 5x 2 3 . 0.
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicated the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work will receive only 1 credit.
15. a. Write the function that is the inverse function of y 5 3x 1 1.
b. Sketch the graph of the function y 5 3x 1 1 and of its inverse.
c. What are the coordinates of the point of intersection of the function
and its inverse?
16. a. Write a recursive definition for the geometric sequence of five terms in
which a1 5 3 and a5 5 30,000.
b. Write the sum of the sequence in part a in sigma notation.
c. Find the sum from part b.
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CHAPTER
7
CHAPTER
TABLE OF CONTENTS
7-1 Laws of Exponents
7-2 Zero and Negative Exponents
7-3 Fractional Exponents
7-4 Exponential Functions and
Their Graphs
7-5 Solving Equations Involving
Exponents
7-6 Solving Exponential Equations
7-7 Applications of Exponential
Functions
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
EXPONENTIAL
FUNCTIONS
The use of exponents to indicate the product of
equal factors evolved through many different notations. Here are some early methods of expressing a
power using an exponent. In many cases, the variable
was not expressed. Each is an example of how the
polynomial 9x4 1 10x3 1 3x2 1 7x 1 4 was written.
Joost Bürgi (1552–1632) used Roman numerals
above the coefficient to indicate an exponent:
IV
III
II
I
9 1 10 1 3 1 7 1 4
Adriaan van Roomen (1561–1615) used
parentheses:
9(4) 1 10(3) 1 3(2) 1 7(1) 1 4
Pierre Hérigone (1580–1643) placed the coefficient before the variable and the exponent after:
9x4 1 10x3 1 3x2 1 7x1 1 4
James Hume used a raised exponent written as a
Roman numeral:
9xiv 1 10xiii 1 3xii 1 7xi 1 4
René Descartes (1596–1650) introduced the symbolism that is in common use today:
9x4 1 10x3 1 3x3 1 7x 1 4
286
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Laws of Exponents
287
7-1 LAWS OF EXPONENTS
When multiplying or dividing algebraic terms, we use the rules for exponents
when the terms involve powers of a variable. For example:
a3b2(3a2b) 5 3a5b3
x4y5 4 x3y2 5 xy3
(c3d2) 2 5 c6d4
These examples illustrate the following rules for exponents. If a and b are
positive integers, then:
Multiplication:
Division:
Raising to a Power:
Power of a Product:
Power of a Quotient:
xa ? xb 5 xab
xa 4 xb 5 xa2b
(x a) b 5 x ab
(xy) a 5 x a ? ya
a
a
A xy B 5 xya
or
xa
xb
5 x a2b (x 0)
In the first example given above, we need the commutative and associative
laws to group powers of the same base in order to apply the rules for exponents.
a3b2(3a2b) 5 3a2a3b2b
2 3
Commutative property
2
5 3(a a )(b b)
Associative property
213 211
5 3a
xa ? xb 5 xa 1 b
b
5 3a5b3
Note: When a variable has no exponent, the exponent is understood to be 1.
In the second example, we use the rule for the division of powers with like
bases:
x4y5
x4y5 4 x3y2 5 x3y2
4
y5
5 xx3 ? y2
5 x423y522
xa
xb
5 xa2b
5 xy3
In the third example, we can use the rule for multiplying powers with like
bases or the rule for raising a power to a power:
(c3d 2) 2
(c3d 2) 2
5 (c3d 2)(c3d2)
x2 5 x ? x
5 (c3) 2(d 2) 2
(xy) a 5 xa ? ya
5 c3c3d 2d 2
Commutative property
5 c3(2)d 2(2)
(xa) b 5 xab
5 c313d 212
xa ? xb 5 xa1b
5 c6d4
5 c6d4
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Exponential Functions
EXAMPLE 1
a4 (3b) 7
Simplify: a(3b) 2
Solution
a4 (3b) 7
a(3b) 2
4
(3b) 7
5 aa1 ? (3b) 2 5 a421(3b) 722 5 a3(3b) 5 5 35a3b5 5 243a3b5 Answer
Alternative
Solution
a4 (3b) 7
a(3b) 2
5 a1 (3) 2(b) 2 5 a421(3) 722 (b) 722 5 a3 (3) 5(b) 5 5 243a3b5 Answer
a4 (3) 7 (b) 7
EXAMPLE 2
(23) 4 (2) 5
What is the value of the fraction (23(2)) 3 ?
Solution
(23) 4 (2) 5
(23(2)) 3
(23) 4 (2) 5
5 (23) 3(2) 35 (23)423(2)523 5 (23)1(2)2 5 (23)(4) 5 212 Answer
Evaluating Powers
Note that the expression (3a)4 is not equivalent to 3a4. We can see this by using
the rule for multiplying powers with like bases or the rule for raising a power to
a power. Let a 5 2 or any number not equal to 0 or 1:
3a4 5 3(a)(a)(a)(a) 5 3(2)(2)(2)(2) 5 48
(3a)4 5 (3a)(3a)(3a)(3a) 5 (3 3 2)(3 3 2)(3 3 2)(3 3 2) 5 (6)(6)(6)(6) 5 1,296
or
4
4
4
4
4
(3a) 5 (3) (a) 5 (3) (2) 5 (81)(16) 5 1,296
Similarly, since 2a 5 21a, the expression 2a4 is not equivalent to (2a)4. Let
a 5 2 (or any number not equal to 0 or 1):
2a4 5 2(a)(a)(a)(a) 5 2(2)(2)(2)(2) 5 216
(2a)4 5 (2a)(2a)(2a)(2a) 5 (22)(22)(22)(22) 5 16
Exercises
Writing About Mathematics
1. Randy said that (2)3(5)2 5 (10)5. Do you agree with Randy? Justify your answer.
2. Natasha said that (2)3(5)3 5 (10)3. Do you agree with Natasha? Justify your answer.
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Zero and Negative Exponents
289
Developing Skills
In 3–26, simplify each expression. In each exercise, all variables are positive.
3. x3 ? x4
4. y ? y5
5. x6 4 x2
6. y4 4 y
7. (x5) 2
8. (2y4) 3
9. 102 ? 104
10. –26 ? 22
11. x4 ? x2y3
12. xy5 ? xy2
13. 2(3x3) 2
14. (23x3) 2
15. x8y6 4 (x3y5)
16. x9y7 4 (x8y7)
17. (x2y3)3 ? (x2y)
18. (22x) 4 ? (2x3) 2
(4x) 3
19. 4x3
x2 (y3z) 3
23. (x2y) 2z
20.
3(x3) 4y5
3x7
2x4y6
21. (2x3y4)
3 5
24. A 2a
a2 B ? b
25.
22.
4(ab) 2c5
abc
A (xy2) 2 B
x3y5
2
(ax) yb
26. axy
Applying Skills
27. What is the value of n if 83 5 2n?
28. What is the value of a if 272 5 9a?
29. If 3a11 5 x and 3a 5 y, express y in terms of x.
30. If 25b11 5 x and 5b 5 y, express x in terms of y.
In 31–33, the formula A 5 P(1 1 r)t expresses the amount A to which P dollars will increase if
invested for t years at a rate of r per year.
31. Find A when P 5 $500, r 5 0.04 and t 5 5.
32. Find the amount to which $2,400 will increase when invested at 5% for 10 years.
33. What is the minimum number of years that $1 must be in invested at 5% to increase to $2?
(Use a calculator to try possible values of t.)
7-2 ZERO AND NEGATIVE EXPONENTS
Zero Exponents
We know that any nonzero number divided by itself is 1:
4
4
51
34
34
5 81
81 5 1
75
75
In general, for x 0 and n a positive integer:
xn
xn
51
51
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Exponential Functions
Can we apply the rule for the division of powers with like bases to these
examples?
4
4
34
34
2
5 222 5 2222 5 20
5 3424 5 30
75
75
5 7525 5 70
In general, for x 0 and n a positive integer:
xn
xn
5 xn2n 5 x0
In order for the rule for the division of powers with like bases to be consistent with ordinary division, we must be able to show that for all x 0, x0 5 1.
Multiplication:
Division:
Raising to a Power:
Power of a Product:
Power of a Quotient:
xn ? x0 5 xn+0 5 xn
xn 4 x0 5 xn20 5 xn
(x0) n 5 x0 ? n 5 x0 5 1
(xy)0 5 x0 ? y0 5 1 ? 1 5 1
0
0
A xy B 5 xy0 5 11 5 1
xn ? x0 5 xn ? 1 5 xn
xn 4 x0 = xn 4 1 5 xn
(x0) n 5 1n 5 1
(xy)0 5 1
0
A xy B 5 1
Therefore, it is reasonable to make the following definition:
DEFINITION
If x 0, x0 5 1.
EXAMPLE 1
2
Find the value of 3x
x2 in two different ways.
3x2
x2
Solution
2
5 3 ? xx2 5 3(1) 5 3
or
3x2
x2
5 3x222 5 3x0 5 3(1) 5 3
Answer 3
Negative Exponents
When using the rule for the division of powers with like bases, we will allow the
exponent of the denominator to be larger than the exponent of the numerator.
34
36
34
34
1
1
1
5 312 3
3 34 5 32 3 34 5 32 3 1 5 32
34
36
5 3426 5 322
5
54
5
5
1
1
1
5 513 3
3 5 5 53 3 5 5 53 3 1 5 53
5
54
5 5124 5 523
xa
xa 1 b
a
a
5 x1b ?? xxa 5 x1b ? xxa 5 x1b ? 1 5 x1b
xa
xa1b
5 xa2(a1b) 5 x2b
This last example suggests that x2b 5 x1b. Before we accept that x2b 5 x1b, we
must show that this is consistent with the rules for powers.
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Zero and Negative Exponents
Multiplication: 25 ? 223 5 251(23) 5 22
25 ? 223 5 25 ? 213 5 22
Division: 62 4 625 5 622(25) 5 67
62 4 625 5 62 4 615 5 62 ? 65 5 67
Raising to a Power: (324) 22 5 324(22) 5 38
(324) 22 5 A 314 B
2 23 4
22
8
5 3128 5 11 5 31 5 38
38
8 212
Power of a Product: (x y ) 5 x y
8
5x ?
1
y12
8
5 yx12
(x2y23) 4 5 (x2 ? y13) 4 5 (x2) 4 A y13 B 5 yx12
4
Power of a Quotient:
A xy5 B
3
A y5 B
x3
22
8
1
26
y10
6
5 yx210 5 x1 5 x6
y10
22
y10
5 x13 2 5 x16 5 x6
A y5 B
y10
Therefore, it is reasonable to make the following definition:
DEFINITION
If x 0, x2n 5 x1n.
EXAMPLE 2
Show that for x 0, x12n 5 xn.
1
x2n
Solution
n
n
5 11 5 11 ? xxn 5 x1 5 xn
n
n
x
x
EXAMPLE 3
4 23
Write aabb22 with only positive exponents.
a4b23
ab22
Solution
3
5 a421b23 2 (22) 5 a3b21 5 a3 ? b1 5 ab
Answer
EXAMPLE 4
Express as a fraction the value of 2x0 1 3x23 for x 5 5.
Solution 2x0 1 3x23 5 2(5)0 1 3(5)23 5 2(1) 1 3 3
Answer
253
125
1
53
3
3
250
5 2 1 125
5 125
1 125
5 253
125
291
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Exponential Functions
Exercises
Writing About Mathematics
1. Kim said that a0 1 a0 5 a010 5 a0 5 1. Do you agree with Kim? Explain why or why not.
2. Tony said that a0 1 a0 5 2a0 5 2. Do you agree with Tony? Explain why or why not?
Developing Skills
In 3–10, write each expression as a rational number without an exponent.
3. 521
4. 422
23
7. A 15 B
8.
A 23 B
5. 622
21
6.
0
9. 4322
10.
A 12 B
21
(2 ? 5) 24
522
In 11–22, find the value of each expression when x 0.
11. 70
15. (4x)0
19. A 34 B
0
12. (25)0
13. x0
14. 240
16. 4x0
17. 22x0
18. (22x)0
30
20. 4
30
3x0
22. (4x)
0
21. 40
In 23–34, evaluate each function for the given value. Be sure to show your work.
23. f(x) 5 x23 ? x4; f(1)
24. f(x) 5 x 1 x25; f(3)
25. f(x) 5 (2x)26 4 x3; f(23)
26. f(x) 5 (x27) 4; f(26)
22
27. f(x) 5 A x1 1 32 B ; f(2)
28. f(x) 5 10x 1 1022x; f(3)
29. f(x) 5 x27 4 x8; f A 34 B
30. f(x) 5 (3x23 2 2x23) 2; f(22)
31. f(x) 5 x8 A x22 1 13 B ; f A 12 B
x
32. f(x) 5 A (2x)
22 B
21
x
33. f(x) 5
34. f(x) 5 4 A 12 B
1
; f(25)
1 1 x221
2x
21
; f(8)
1 3 A 12 B
2x
; f(3)
In 35–63, write each expression with only positive exponents and express the answer in simplest
form. The variables are not equal to zero.
35. x24
36. a26
37. y25
38. 2x22
39. 7a24
40. –5y28
41. (2x)22
42. (3a)24
43. (4y)23
44. 2(2x)22
45. 2(3a)24
46. (22x)22
47. x123
48. y127
49. a325
50. a624
2
51. 9x
a23
52.
(2x) 25
x23
(2a) 21
53. 2(a) 22
4y23
54. 2y21
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Fractional Exponents
23 23
57. A 3m
2n22 B
55. (xy5z22) 21
56. (a5b25c24) 23
58. A 3x6ab
23 24 B
y
59. x24y22
60. (3ab22)(3a21b3)
3v4 21
62. A 249u
27u4v7 B
63. x21 1 x25
4
x5y24
21
4a22
61. 264x
2x5b24
293
In 64–75, write each quotient as a product without a denominator. The variables are not equal to
zero.
64. (xy) 4 (xy3)
65. (a2b3) 4 (ab5)
66. x3 4 (x3y 4)
67. 12ab 4 2ab2
68. a123
69. x34
70. 4a8 3
71. 9x3625
0 23
72. a3a21bb23
20x0y25
15x22y2
73. 4x21y5
5b23
75. 25a
50a21b
74. 3xy5
76. Find the value of a0 1 (4a)21 1 4a22 if a 5 2.
77. Find the value of (25a)0 2 5a22 if a 5 3.
78. If A 34 B
23
2
a
1 A 83 B 5 32b, find the values of a and of b.
3
79. Show that 3 3 1022 5 100
.
7-3 FRACTIONAL EXPONENTS
We know the following:
Since 32 5 9:
Since 53 5 125:
Since 24 5 16:
In general, for x . 0:
!9 5 3 or "32
3
3 3
!
125 5 5 or "
5
4
4 4
!
16 5 2 or "
2
n n
"
x 5x
In other words, raising a positive number to the nth power and taking the
nth root of the power are inverse operations. Is it possible to express the nth
root as a power? Consider the following example:
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Exponential Functions
!x ? !x 5 x
(1) Let x be any positive real number:
(2) Assume that !x can be expressed as xa:
xa ? xa 5 x
xa1a 5 x1
(3) Use the rule for multiplying powers with like bases:
x2a 5 x1
2a 5 1
a 5 12
1
(4) Therefore, we can write the following equality:
(5) Since for x . 0, !x is one of the two equal factors
of x, then:
1
x2 ? x2 5 x
!x 5 x2
1
We can apply this reasoning to any power and the corresponding root:
y
n
n
n
n
!
x? !
x ? !
x ? c? !
x5x
n factors
Assume that "x can be expressed as xa:
n
y
xa ? xa ? xa ? xa ? c ? xa 5 x
n factors
(xa) n 5 x1
an 5 1
a 5 n1
n
For n a positive integer, !
x is one of the n equal factors of x, and xn is one
of the n equal factors of x. Therefore:
1
n
!
x 5 xn
1
Before we accept this relationship, we must verify that operations with radicals and with fractional exponents give the same results.
Multiplication: !8 3 !2 5 !16 5 4
1
1
1
1
(8) 2 3 (2) 2 5 (23) 2 3 (2) 2
3
1
5 (2) 2 3 (2) 2
3
1
5 (2) 212
4
5 (2) 2
5 (2) 2 5 4
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Fractional Exponents
295
5
5
5 !5
3 !5
5 5 !5
Division: !5
5 5 !5
!5
51
1
52
1
1
5 51 2 2 5 5 2
3
3
Raising to a Power: "26 5 ! 64 5 4
(26) 3 5 26 A3 B 5 22 5 4
1
1
Verifying that the laws for power of a product and power of a quotient also
hold is left to the student. (See Exercise 83.) These examples indicate that it is
reasonable to make the following definition:
DEFINITION
n
If x 0 and n is a positive integer, !
x 5 xn.
1
From this definition and the rules for finding the power of a power and the
definition of negative powers, we can also write the following:
n m
n
"
x 5 A!
x Bm 5 x n
m
1
m
"x
n
5
1
n
m
A !xB
m
5 x2n
Note: When x , 0, these formulas may lead to invalid results.
3
For instance: (28) 3 5 !28 5 22
1
1
3
and
6
6
(28) 6 5 "
(28) 2 5 !
64 5 2
5 26
2
However:
1
2
Thus, (28) 3 2 (28) 6 even though the exponents are equal.
A calculator will evaluate powers with fractional exponents. For instance, to
2
find (125) 3:
ENTER: 125
^
DISPLAY:
>
14411C07.pgs
2 3
(
)
ENTER
125 (2/3)
25
3
3
We can also evaluate (125) 3 by writing it as "
1252 or as A "
125 B .
2
ENTER:
MATH
)
DISPLAY:
4
2
ENTER:
125 x 2
(
x2
ENTER
DISPLAY:
3√ ( 1 2 5 2)
MATH
4
125
)
ENTER
( 3√ ( 1 2 5 ) ) 2
25
25
3
3
The calculator shows that (125) 3 5 "1252 5 A "125 B .
2
2
)
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Exponential Functions
EXAMPLE 1
3
Find the value of 8124.
3
8124 5 1 3 5
Solution
814
Calculator ENTER: 81 ^
Solution
DISPLAY:
(-)
(
1
A !81 B
4
3 4
)
3
1
Answer
5 1 3 5 27
(3)
MATH
ENTER
ENTER
>
14411C07.pgs
81 -(3/4)Frac
1/27
Note that we used the Frac function from the MATH menu to convert the
answer to a fraction.
EXAMPLE 2
1
Find the value of 2a0 2 (2a)0 1 a3 if a 5 64.
1
1
2a0 2 (2a)0 1 a3 5 2(64)0 2 [2(64)]0 1 643
Solution
3
5 2(1) 2 1 1 !
64
522114
5 5 Answer
Exercises
Writing About Mathematics
n 0
1. Use exponents to show that for a . 0, A !
aB 5 1.
4
2. Use exponents to show that for a . 0, " !a 5 !
a.
Developing Skills
In 3–37, express each power as a rational number in simplest form.
1
1
3. 42
1
4. 92
1
7. 1253
5. 1002
1
8. 2163
1
9. 325
1
6. 83
1
10. (3 3 12) 2
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297
Fractional Exponents
1
1
11. (2 3 8) 4
5
3
16. 273
1
3
1
1
2
1
35. 2f (3)
22
1
2
1
1
3
1
26. 4 3 42
5
2
1
36. A 2.3 3 10
1 (4) g
5
7
2
29. 123 4 123
1
213
B A 5.2 3 10 B
223
1
30. 33 4 33
1
33. 40 1 422
32. 1253 4 1253
22 212
22. 12523
25. 74 3 74
28. 24 3 84
31. 43 4 46
2
21. 10022
24. 53 3 53
27. 32 3 325
18. 325
3
20. 823
23. 32 3 32
4
17. 10,0004
1
19. 922
14. 492
13. 24(1,000) 3
4
15. 83
3
1
12. 5(81) 4
34. 922 1 92
A 2(3) 2 1 3122 B 3
2
37.
6 A 2 1 14 B 22
1
In 38–57, write each radical expression as a power with positive exponents and express the answer
in simplest form. The variables are positive numbers.
38. !7
39. !6
5 3
43. "
2
4
42. !
3
46. !25a
50. "9a22b6
54.
3
40. !
12
1
5
5
"xyz
3
41. !
15
5
44. A !
9B 4
45.
1
3
A !5B
47. "49x2
48. "64a3b6
49. 12"18a6b2
55. #9a
4b4
56. #w y5x
8 4
4
57. #!a ? "b7
3
52. "
27a3
3a
51. #4b
22
10
15 20
4
53. "
64x5
In 58–73, write each power as a radical expression in simplest form. The variables are positive
numbers.
1
1
58. 32
3
5
62. 52
1
1
(x5y6) 7
237
z
1
67. (25x2y) 2
61. 93
5
65. 13
43
52
1
1
68. (50ab4) 2
1 2
3 3
71. 5 a1
1
60. 63
64. 62
63. 124
66. (x13) 7
70.
1
59. 52
72.
69. (16a5b6) 4
5
A 232x
y4 B
10
1
1 5 3
6 6 6
73. 8 a b4 1
(27c ) 6
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Exponential Functions
In 74–82, write each expression as a power with positive exponents in simplest form.
1
2
74. ¢ 2a1 ≤
75. A 3x4b2 B 3
6
x2y
3a6
z
3
77. A 27x8a
9 24 21 B
a z
2 6
4a b 2
76. A 25a
21 B
b
4 6
2
6 5
a
79. "
5 5
78. "x2y ? "x4y3
1
"11x5y4
80.
5 2
"2x y
"a
"48xy2
81. 3 2 4
"6x y
3
1
4
82. A "2xy2 B A "
16x2y B
5
83. Verify that the laws for power of a product and power of a quotient are true for the following examples. In each example, evaluate the left side using the rules for radicals and the
right side using the rules for fractional exponents:
3
3
a. A !
27 ? !
3 B 2 5 A273 ? 33 B 2
?
1
1
1
3
3 ?
2
b. A !3
5 ¢ 31 ≤
!9 B
92
7-4 EXPONENTIAL FUNCTIONS AND THEIR GRAPHS
Two friends decide to start a chess club. At the first meeting, they are the only
members of the club. To increase membership, they decide that at the next meeting, each will bring a friend. At the second meeting there are twice as many, or
2 3 2 5 22, members. Since they still want to increase membership, at the third
meeting, each member brings a friend so that there are 2 3 22 5 23 members.
If this pattern continues, then at the nth meeting, there will be 2n members.
We say that the membership has increased exponentially or is an example of
exponential growth.
Often quantities that change exponentially decrease in size. For example,
radioactive elements decrease by an amount that is proportional to the size of
the sample. This is an example of exponential decay. The half-life of an element
is the length of time required for the amount of a sample to decrease by onehalf. If the weight of a sample is 1 gram and the half life is t years, then:
• After one period of t years, the amount present is (1) 12 5 221 grams.
• After two periods of t years, the amount present is A 12 B A 12 B 5 14 5 222
grams.
• After three periods of t years, the amount present is A 14 B A 12 B 5 18 5 223
grams.
• After n periods of t years, the amount present is 21n 5 22n grams.
Each of the examples given above suggests a function of the form f(x) 5 2x.
To study the function f(x) 5 2x, choose values of x, find the corresponding value
of f(x), and plot the points. Draw a smooth curve through these points to represent the function.
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Exponential Functions and Their Graphs
x
2x
–3
223 5 13
–2
–1
2
22
2 5 12
2
21
2 5 11
2
x
2x
1
8
1
21 5 2
2
1
4
3
2
22 5 "23
!8 < 2.8
1
2
2
22 5 4
4
y
3
0
20 5 1
1
5
2
!2 < 1.4
!32 5.7
1
2
22 5 !2
22 5 "25
3
23 5 8
8
1
y
y
5
y 2x
1
x
1
O
The function y 5 2x is an increasing function. As we trace the graph from
left to right, the values of y increase, that is, as x increases, y also increases.
If we reflect the graph of y 5 2x over the y-axis, (x, y) → (2x, y) and the
x
equation of the image is y 5 22x , that is, y 5 21x or y 5 A 12 B . Compare the graph
x
shown below with the values of A 12 B shown in the table.
x
23
22
21
0
A 12 B
A 12 B
A 12 B
A 12 B
A 12 B
x
y
x
23
5 23
8
1
22
5 22
4
2
21
5 21
2
3
1
4
0
A 12 B
A 12 B
A 12 B
A 12 B
A 12 B
x
y
y
1
1
2
2
1
4
3
1
8
4
1
16
y A 12 B
x
1
O
1
x
Note that the function y 5 A 12 B is a decreasing function. As we trace the
graph from left to right, the values of y decrease, that is, as x increases, y
decreases.
An exponential function is a function of the form y 5 bx where b is a positive number not equal to 1. If b . 1, the function is an increasing function. If
0 , b , 1, the function is a decreasing function.
We can use a graphing calculator to draw several exponential functions with
x
b . 1. For example, let y 5 A 52 B , y 5 3x, and y 5 5x. First, set the WINDOW so
that Xmin 5 23, Xmax 5 3, Ymin 5 21, and Ymax 5 10.
x
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Exponential Functions
ENTER:
Y
5 2
(
)
3 ^
X,T,,n
ENTER
5 ^
X,T,,n
GRAPH
DISPLAY:
y 5x
X,T,,n
^
ENTER
y 3x
y A 52 B
x
If we press TRACE , the calculator will display the values x 5 0, y 5 1 for
the first graph. By pressing , the calculator will display this same set of
values for the second function and again for the third. The point (0, 1) is a point
on every function of the form y 5 bx. For each function, as x decreases through
negative values, the values of y get smaller and smaller but are always positive.
We say that as x approaches 2`, y approaches 0. The x-axis or the line y 5 0 is
a horizontal asymptote.
EXAMPLE 1
x
a. Sketch the graph of y 5 A 32 B .
1.2
b. From the graph, estimate the value of A 32 B , the value of y when x 5 1.2.
x
c. Sketch the graph of the image of the graph of y 5 A 32 B under a reflection in
the y-axis.
d. Write an equation of the graph drawn in part c.
Solution a.
x
22
21
0
1
2
3
y
22
2
A 32 B 5 A 23 B
21
1
A 32 B 5 A 23 B
0
A 32 B 5 1
1
A 32 B 5 32
2
A 32 B 5 94
3
A 32 B 5 278
y
5 49
5 23
yA
B
3x
2
(1.2, 1.6)
1
1 O
y A 32 B
x
x
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Exponential Functions and Their Graphs
b. Locate the point 1.2 on the x-axis. Draw a vertical line to the graph. From
that point on the graph draw a horizontal line to the y-axis. Read the
approximate value of y from the graph. The value is approximately 1.6.
x
c. For each point (a, b) on the graph of y 5 A 32 B , locate the point (2a, b) on
the graph of the image.
2x
x
d. The equation of the image is y 5 A 32 B or y 5 A 23 B .
Answers a. Graph
b. 1.6
c. Graph
x
d. y 5 A 23 B
We can use the graphing calculator to find the values of the function used
x
to graph y 5 A 32 B in Example 1.
STEP 1. Enter the function into Y1. Then
Normal S c i E n g
Float 0 1 2 3 4 5 6 7 8 9
Radian D e g r e e
Func
Par Pol seq
Connected D o t
Sequential S i m u l
Real a+bi r e u i
F u l l H o r i z G-T
change the graphing mode to split
screen mode (G-T mode) by pressing
MODE and selecting G-T. This will
display the graph of the function
alongside a table of values.
>
14411C07.pgs
STEP 2. Press
2nd
TBLSET to enter the
TABLE SETUP screen. Change
TblStart to 22 and make sure that
Tbl is set to 1. This tells the calculator to start the table of values at 22
and increase each successive x-value
by 1 unit.
STEP 3. Finally, press
TABLE SETUP
TblStart=-2
Tbl=1
Indpnt: Auto Ask
Depend: Auto Ask
to plot the
graph and the table of values. Press
2nd
TABLE
to use the table.
Move through the table of values
by pressing the up and down arrow
keys.
GRAPH
X
-2
1
0
1
2
3
4
X=-2
Y1
.4444
.6667
1
1.5
2.25
3.375
5.063
By tracing values on the graph of y 5 A 32 B , we can again see that as x
approaches 2`, y approaches 0, and as x approaches `, y approaches `.
x
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Exponential Functions
The Graph of y 5 ex
In Section 6-7, we introduced the irrational number e. For many real-world
applications, the number e is a convenient choice to use as the base. This is why
the number e is also called the natural base, and an exponential function with e
as the base is called a natural exponential function. Recall that
e 5 2.718281828 . . .
Note that e is an irrational number. The above pattern does not continue to
repeat.
The figure on the left is a graph of the natural exponential function y 5 ex.
The figure on the right is a calculator screen of the same graph along with a
table of selected values.
X
-3
-2
-1
0
1
2
3
X=-3
Y1
.04979
.13534
.36788
1
2.7183
7.3891
20.086
Exercises
Writing About Mathematics
1. Explain why (0, 1) is a point on the graph of every function of the form y 5 bx.
2. Explain why y 5 bx is not an exponential function for b 5 1.
Developing Skills
In 3–6: a. Sketch the graph of each function. b. On the same set of axes, sketch the graph of the image
of the reflection in the y-axis of the graph drawn in part a. c. Write an equation of the graph of the
function drawn in part b.
3. y 5 4x
4. y 5 3x
x
7. a. Sketch the graph of y 5 A 53 B .
x
5. y 5 A 72 B
x
6. y 5 A 34 B
x
b. From the graph of y 5 A 53 B , estimate the value of y, to the nearest tenth, when x 5 2.2.
x
c. From the graph of y 5 A 53 B , estimate the value of x, to the nearest tenth, when y 5 2.9.
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Exponential Functions and Their Graphs
303
8. a. Sketch the graph of f(x) 5 2x.
b. Sketch the graph of the image of f(x) 5 2x under a reflection in the x-axis.
c. Write an equation for the function whose graph was sketched in part b.
9. a. Sketch the graph of f(x) 5 1.2x.
b. Sketch the graph of the image of f(x) 5 1.2x under a reflection in the x-axis.
c. Write an equation for the function whose graph was sketched in part b.
10. a. Make a table of values for ex for integral values of x from 22 to 3.
b. Sketch the graph of f(x) 5 ex by plotting points and joining them with a smooth curve:
1
c. From the graph, estimate the value of e2 and compare your answer to the value given by
a calculator.
Applying Skills
11. The population of the United States can be modeled by the function p(x) 5 80.21e0.131x
where x is the number of decades (ten year periods) since 1900 and p(x) is the population in
millions.
a. Graph p(x) over the interval 0 x 15.
b. If the population of the United States continues to grow at this rate, predict the population in the years 2010 and 2020.
12. In 1986, the worst nuclear power plant accident in history occurred in the Chernobyl
Nuclear Power Plant located in the Ukraine. On April 26, one of the reactors exploded,
releasing large amounts of radioactive isotopes into the atmosphere. The amount of plutonium present after t years can be modeled by the function:
y 5 Pe20.0000288t
where P represents the amount of plutonium that is released.
a. Graph this function over the interval 0 t 100,000 and P 5 10 grams.
b. If 10 grams of the isotope plutonium-239 were released into the air, to the nearest hundredth, how many grams will be left after 10 years? After 100 years?
c. Using the graph, approximate how long it will take for the 10 grams of plutonium-239 to
decay to 1 gram.
13. a. Graph the functions y 5 x4 and y 5 4x on a graphing calculator using the following viewing windows:
(1) Xmin 5 0, Xmax 5 3, Ymin 5 0, Ymax 5 50
(2) Xmin 5 0, Xmax 5 5, Ymin 5 0, Ymax 5 500
(3) Xmin 5 0, Xmax 5 5, Ymin 5 0, Ymax 5 1,000
b. How many points of intersection can you find? Find the coordinates of these intersection
points to the nearest tenth.
c. Which function grows more rapidly for increasing values of x?
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Exponential Functions
7-5 SOLVING EQUATIONS INVOLVING EXPONENTS
We know that to raise a power to a power, we multiply exponents. Therefore, for
positive values of x and non-zero integer values of a:
Ax a B 5 x a(a) 5 x1 5 x
(x a) a 5 x aAa B 5 x1 5 x
1
1
1
a
1
2
We can use this relationship to solve for x in an equation such as x3 5 25. To
2
solve for x, we need to raise x3 to the power that is the reciprocal of the exponent 32. The reciprocal of 32 is 32.
2
x3 5 25
A x3 B 2 5 252
2
3
3
3
x1 5 252
3
x 5 252
Note that 252 means A252 B , that is, the cube of the square root of 25.
3
1
3
x 5 A !25B 3 5 53 5 125
EXAMPLE 1
Solve each equation and check: a. 2a23 2 1 5 15
Solution
How to Proceed
(1) Write the equation
with only the variable
term on one side of the
equation:
(2) Divide both sides of the
equation by the coefficient
of the variable term:
(3) Raise both sides of the
equation to the power
that is the reciprocal of
the exponent of the
variable:
(4) Simplify the right side
of the equation:
3 5
b. 2"
x 1 1 5 487
a. 2a23 2 1 5 15
3 5
2"
x 5 487
2a23 5 16
5
a23 5 8
1
3 5
b. 2"
x 1 1 5 487
x3 5 243
1
(a23) 23 5 823
1
Ax3 B 5 5 2435
5 3
3
3
a 5 823
x 5 2435
a 5 11
x 5 2435
5
5
83
1
3
!8
1
2
3
5
5 A!
243B 3
5 33
5 27
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Solving Equations Involving Exponents
(5) Check the solution:
a.
2a23 2 1 5 15
2 A 12 B
23
?
– 1 5 15
?
2 (2)3 2 1 5 15
?
2(8) 2 1 5 15
b.
2"x5 1 1 5 487
3
3
2" 275 1 1 5 487
?
3
2"(33) 5 1 1 5 487
?
2"315 1 1 5 486
3
?
?
2(35) 1 1 5 487
?
16 2 1 5 15
487 5 487 ✔
15 5 15 ✔
Answers a. a 5
1
2
305
b. x 5 27
Exercises
Writing About Mathematics
1
1. Ethan said that to solve the equation (x 1 3) 2 5 5, the first step should be to square both
sides of the equation. Do you agree with Ethan? Explain why or why not.
1
2. Chloe changed the equation a22 5 36 to the equation a12 5 36
and then took the square root
of each side. Will Chloe’s solution be correct? Explain why or why not.
Developing Skills
In 3–17 solve each equation and check.
1
1
3. x3 5 4
3
6. b2 5 8
2
4. a5 5 2
5. x5 5 9
7. x22 5 9
1
8. b25 5 32
3
10. 9a24 5 13
9. 2y21 5 12
1
1
12. 5x2 1 7 5 22
13. 14 2 4b3 5 2
15. 3a3 5 81
16. x5 5 3,125
3
11. 5x4 5 40
1
14. (2x) 2 1 3 5 15
17. z2 5 !81
1
In 18–23, solve for the variable in each equation. Express the solution to the nearest hundredth.
2
18. x23 5 24
21. 3z3 1 2 5 27
3
19. y3 5 6
20. a24 5 0.85
22. 5 1 b5 5 56
23. (3w) 9 1 2 5 81
1
3
2
3
24. Solve for x and check: x 5 10. Use the rule for the division of powers with like bases to
x
simplify the left side of the equation.
Applying Skills
3
25. Show that if the area of one face of a cube is B, the volume of the cube is B2.
26. If the area of one face of a cube is B and the volume of the cube is V, express B in terms of V.
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7-6 SOLVING EXPONENTIAL EQUATIONS
Solving Exponential Equations With the Same Base
An exponential equation is an equation that contains a power with a variable
exponent. For example, 22x 5 8 and 5x21 5 0.04 are exponential equations.
An exponential function y 5 bx is a one-to-one function since it is increasing for b . 1 and decreasing for 0 , b , 1. Let y1 5 bx1 and y2 5 bx2. If y1 5 y2,
then bx1 5 bx2 and x1 5 x2.
In general, if b p 5 b q, then p 5 q.
We can use this fact to solve exponential equations that have the same base.
EXAMPLE 1
Solve and check: 3x 5 32x22
Solution Since the bases are equal, the exponents must be equal.
3x 5 32x22
x 5 2x 2 2
2x 5 22
x52
Check
3 5 32x22
x
?
32 5 32(2)22
32 5 32 ✔
Answer x 5 2
Solving Exponential Equations With Different Bases
How do we solve exponential equations such as 22x 5 8 or 5x21 5 0.04? One
approach is, if possible, to write each term as a power of the same base. For
example:
22x 5 8
5x21 5 0.04
22x 5 23
4
5x21 5 100
2x 5 3
1
5x21 5 25
x 5 32
5x21 5 512
5x21 5 522
x 2 1 5 22
x 5 21
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Solving Exponential Equations
307
EXAMPLE 2
Solve and check: 4a 5 8a11
Solution The bases, 4 and 8, can each be written as a power of 2: 4 5 22, 8 5 23.
4a 5 8a11
2 a
Check
4a 5 8a11
?
423 5 82311
?
423 5 822
3 a11
(2 ) 5 (2 )
22a 5 23a13
2a 5 3a 1 3
1 ? 1
43 5 82
1
1
64 5 64
2a 5 3
a 5 23
✔
Answer a 5 23
EXAMPLE 3
Solve and check: 3 1 7x21 5 10
Solution Add 23 to each side of the equation to isolate the power.
3 1 7x21 5 10
Check
3 1 7x21 5 10
?
3 1 7221 5 10
?
3 1 71 5 10
10 5 10 ✔
7x21 5 7
x2151
x52
Answer x 5 2
Exercises
Writing About Mathematics
1. What value of a makes the equation 6a 5 1 true? Justify your answer.
2. Explain why the equation 3a 5 5a21 cannot be solved using the procedure used in this section.
Developing Skills
In 3–14, write each number as a power.
3. 9
4. 27
5. 25
7. 1,000
8. 32
9. 81
11. 0.001
12. 0.125
13. 0.81
6. 49
1
10. 216
14. 0.16
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Exponential Functions
In 15–38, solve each equation and check.
15. 2x 5 16
16. 3x 5 27
17. 5x 5 51
1
18. 7x 5 49
19. 4x12 5 42x
20. 3x11 5 32x13
21. 63x 5 6x21
22. 3x12 5 9x
23. 25x 5 5x13
24. 49x 5 73x11
25. 22x11 5 16x
26. 9x21 5 27x
27. 100x 5 1,000x21
28. 125x21 5 25x
x
30. A 14 B 5 812x
1
29. 622x 5 A 36
B
x
31. A 13 B 5 912x
33. (0.25)x22 5 4x
36. 5 1 7x 5 6
2
32. (0.01)2x 5 10022x
34. 5x21 5 (0.04)2x
35. 4x 1 7 5 15
37. e2x 1 2 5 ex 2 1
38. 3x 1 2 5 36
2
7-7 APPLICATIONS OF EXPONENTIAL FUNCTIONS
There are many situations in which an initial value A0 is increased or decreased
by adding or subtracting a percentage of A0. For example, the value of an investment is the amount of money invested plus the interest, a percentage of the
investment.
Let An be the value of the investment after n years if A0 dollars are invested
at rate r per year.
• After 1 year:
A1 = A0 1 A0r 5 A0(1 1 r)
A1 5 A0(1 1 r)
• After 2 years:
A2 5 A1 1 A1r 5 A0(1 1 r) 1 A0(1 1 r)r
5 A0(1 1 r)(1 1 r) 5 A0(1 1 r)2
A2 5 A0(1 1 r)2
• After 3 years:
A3 5 A2 1 A2r 5 A0(1 1 r)2 1 A0(1 1 r)2r
5 A0(1 1 r)2(1 1 r) 5 A0(1 1 r)3
A3 5 A0(1 1 r)3
• After 4 years:
A4 5 A3 1 A3r 5 A0(1 1 r)3 1 A0(1 1 r)3r
5 A0(1 1 r)3(1 1 r) 5 A0(1 1 r)4
A4 5 A0(1 1 r)4
We see that a pattern has been established. The values at the end of each year
form a geometric sequence with the common ratio (1 1 r).
• After t years:
At 5 At–1 1 At–1r 5 A0(1 1 r)t21 1 A0(1 1 r)t21r 5 A0(1 1 r)t21(1 1 r) 5 A0(1 1 r)t
or
A 5 A0(1 1 r) t
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Applications of Exponential Functions
A calculator can be used to display the value of an investment year by year.
The value of the investment after each year is (1 1 r) times the value of the
investment from the previous year. If $100 is invested at a yearly rate of 5%, the
value of the investment at the end of each of the first 5 years can be found by
multiplying the value from the previous year by (1 1 0.05) or 1.05.
ENTER: 100
DISPLAY:
1.05 ENTER
1.05 ENTER
1.05 ENTER
1.05 ENTER
1.05 ENTER
100*1.05
105
Ans*1.05
110.25
115.7625
121.550625
127.6281563
The value of the investment for each of the first five years is $105.00, $110.25,
$115.76, $121.55, and $127.63.
EXAMPLE 1
Amanda won $10,000 and decided to use it as a “vacation fund.” Each summer,
she withdraws an amount of money that reduces the value of the fund by 7.5%
from the previous summer. How much will the fund be worth after the tenth
withdrawal?
Solution Use the formula A 5 A0(1 1 r) t with A0 5 10,000, r 5 20.075 and t 5 10.
A 5 10,000(1 2 0.075)10 5 10,000(0.925)10
ENTER:
10000 0.925 ^
DISPLAY:
10 ENTER
>
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10000*0.925 10
4585.823414
Answer $4,585.82
Calculator variables can be used to quickly evaluate an exponential model
for different time periods. For instance, to find the value of the fund from
Example 1 after the second and fifth withdrawals:
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Exponential Functions
STEP 1. Let Y1 5 10,000(0.925)x.
STEP 2. Exit the Y5 screen and let X 5 2.
ENTER: 2 STO佡
X,T,,n
2 >X
ENTER
2
Y1
STEP 3. Find the value of Y1 when X 5 2.
ENTER: VARS
ENTER
8556.25
5 >X
ENTER
5
Y1
ENTER
6771.870801
STEP 4. Repeat steps 2 and 3 to find the value of
Y1 when X 5 5.
The value of the fund after the second withdrawal is $8,556.25 and after the
fifth withdrawal, $6,771.87.
Other Types of Compounding Periods
The exponential function A 5 A0 (1 1 r) t, when applied to investments, represents interest compounded annually since interest is added once at the end
of each year. However, this formula can be applied to any increase that takes
place at a fixed rate for any specified period of time. For example, if an annual
5
interest rate of 5% is paid monthly, then interest of 12
% is paid 12 times in one
5
year. Or if an annual interest rate of 5% is paid daily, then interest of 365
% is
paid 365 times in one year. Compare the value of an investment of $100 for
these three cases.
Paid yearly for 1 year: A 5 100(1 1 0.05)1 5 105 or $105.00
Paid monthly for 1 year: A 5 100 A 1 1 0.05
12 B
12
105.1161898 or $105.12
Paid daily for 1 year: A 5 100 A 1 1 0.05
105.1267496 or $105.13
365 B
In general, if interest is compounded n times in one time period and the
number of time periods is t, then:
A 5 A0 A 1 1 nr B
365
nt
Note that as the number of times that the interest is compounded increases,
the value of the investment for a fixed period of time increases.
What happens as we let n, the number of compoundings, increase without
limit? Let nr 5 k1 . Then n 5 rk.
nt
A 5 A0 A 1 1 nr B
A 5 A0 A 1 1 k1 B
rkt
A 5 A0 S A 1 1 k1 B T
k rt
Substitute nr 5 k1 and n 5 rk.
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Applications of Exponential Functions
As n approaches infinity, k approaches infin-
ity. It can be shown that A 1 1
1 k
kB
k
approaches e.
The table on the right shows values of A 1 1 k1 B
rounded to eight decimal places for different
k
values of k. Recall that e is an irrational number
that is the sum of an infinite series:
1
1
1
1 1 1!
1 2!
1 3!
1 c 2.718281828 . . .
Therefore, for very large values of n:
A 5 A0 A 1 1 nr B
nt
311
A 1 1 1k B
k
1
2
10
2.59374246
100
2.70481383
1,000
2.71692393
100,000
2.71826823
1,000,000
2.71828047
5 A0 Q A 1 1 k1 B R 5 A0ert
k rt
This formula can be applied to any change that takes place continuously at
a fixed rate. Large populations of people or of animals can be said to increase
continuously. If this happens at a fixed rate per year, then the size of the population in the future can be predicted.
EXAMPLE 2
In a state park, the deer population was estimated to be 2,000 and increasing
continuously at a rate of 4% per year. If the increase continues at that rate, what
is the expected deer population in 10 years?
Solution Use the formula A 5 A0ert. Let A0 5 2,000, r 5 0.04, and t 5 10.
A 5 2,000e0.04(10) 5 2,000e0.4
ENTER: 2000
DISPLAY:
2nd
ex
0.4
)
ENTER
>
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2000*e (0.4)
2983.649395
Answer The expected population is 2,984 deer.
Note: Since the answer in Example 2 is an approximation, it is probably more
realistic to say that the expected population is 3,000 deer.
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Exponential Functions
The formula A 5 A0ert can also be used to study radioactive decay. Decay
takes place continuously. The decay constant for a radioactive substance is negative to indicate a decrease.
EXAMPLE 3
The decay constant of radium is 20.0004 per year. How many grams will remain
of a 50-gram sample of radium after 20 years?
Solution Since change takes place continuously, use the formula A 5 A0ert. Let
A0 5 50, r 5 20.0004 and t 5 20.
A 5 50e20.0004(20) 5 50e20.008
ENTER: 50
DISPLAY:
ex
2nd
20.008
)
ENTER
>
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50*e (-0.008)
49.60159574
Answer Approximately 49.6 grams
Exercises
Writing About Mathematics
1. Show that the formula A 5 A0(1 1 r)n is equivalent to A 5 A0(2)n when r 5 100%.
2. Explain why, if an investment is earning interest at a rate of 5% per year, the investment is
worth more if the interest is compounded daily rather than if it is compounded yearly.
Developing Skills
In 3–10, find the value of x to the nearest hundredth.
3. x 5 e2
4. x 5 e1.5
5. x = e21
7. 12x 5 e
8. ex3 5 e22
9. x 5 e3e5
In 11–16, use the formula A 5 A0 A 1 1 nr B
11. A0 5 50, r 5 2%, n 5 12, t 5 1
nt
6. xe3 5 e4
10. x 5 e3 1 e5
to find the missing variable to the nearest hundredth.
12. A 5 400, r 5 5%, n 5 4, t 5 3
13. A 5 100, A0 5 25, n 5 1, t 5 2
14. A 5 25, A0 5 200, r 5 250%, n 5 1
15. A 5 250, A0 5 10, n 5 3, t 5 1
16. A 5 6, A0 5 36, n 5 1, t 5 4
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Applications of Exponential Functions
313
Applying Skills
17. A bank offers certificates of deposit with variable compounding periods.
a. Joe invested $1,000 at 6% per year compounded yearly. Find the values of Joe’s investment at the end of each of the first five years.
b. Sue invested $1,000 at 6% per year compounded monthly. Find the values of Sue’s investment at the end of each of the first five years.
c. Who had more money after the end of the fifth year?
d. The annual percentage yield (APY) is the amount an investment actually increases during
one year. Find the APY for Joe and Sue’s certificates of deposit. Is the APY of each
investment equal to 6%?
18. a. When Kyle was born, his grandparents invested $5,000 in a college fund that paid 4% per
year, compounded yearly. What was the value of this investment when Kyle was ready for
college at age 18? (Note that r 5 0.04.)
b. If Kyle’s grandparents had invested the $5,000 in a fund that paid 4% compounded continuously, what would have been the value of the fund after 18 years?
19. A trust fund of $2.5 million was donated to a charitable organization. Once each year the
organization spends 2% of the value of the fund so that the fund decreases by 2%. What
will be the value of the fund after 25 years?
20. The decay constant of a radioactive element is 20.533 per minute. If a sample of the element weighs 50 grams, what will be its weight after 2 minutes?
21. The population of a small town decreased continually by 2% each year. If the population of
the town is now 37,000, what will be the population 8 years from now if this trend continues?
22. A piece of property was valued at $50,000 at the end of 1990. Property values in the city
where this land is located increase by 10% each year. The value of the land increases continuously. What is the property worth at the end of 2010?
23. A sample of a radioactive substance decreases continually at a rate of 20.04. If the weight
of the sample is now 40 grams, what will be its weight in 7 days?
24. The number of wolves in a wildlife preserve is estimated to have increased continually by
3% per year. If the population is now estimated at 5,400 wolves, how many were present 10
years ago?
25. The amount of a certain medicine present in the bloodstream decreases at a rate of 10%
per hour.
a. Which is a better model to use for this scenario: A 5 A0 (1 1 r) t or A 5 A0ert? Explain
your answer.
b. Using both models, find the amount of medicine in the bloodstream after 10.5 hours if
the initial dose was 200 milligrams.
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Exponential Functions
CHAPTER SUMMARY
The rules for operations with powers with like bases can be extended to
include zero, negative, and fractional exponents. If x . 0 and y . 0:
Multiplication:
Division:
Raising to a Power:
Power of a Product:
Power of a Quotient:
xa ? xb 5 xa1b
xa 4 xb 5 xa2b
(x a) b 5 x ab
(xy) a 5 x a ? ya
a
a
A xy B 5 xya
or
xa
xb
5 x a 1 b (x 0)
If x . 0 and n is a positive integer:
n
!
x 5 xn
1
In general, for x . 0 and m and n positive integers:
n m
n
"
x 5 A!
xB m 5 x n
m
1
m
"x
n
5
The graph of the exponential function, y 5 bx:
y
1
n
m
A !xB
m
5 x2 n
y
O
x
O
for b . 1
x
for 0 , b , 1
a
To solve an equation of the form xb 5 c, raise each side of the equation to
b
the a power so that the left member is x1 or x.
An exponential equation is an equation that contains a power with a variable exponent. To solve an exponential equation, write each member with the
same base and equate exponents.
The number e is also called the natural base and the function y 5 ex is called
the natural exponential function.
If a quantity A0 is increased or decreased by a rate r per interval of time, its
value A after t intervals of time is A 5 A0(1 1 r) t.
If a quantity A0 is increased or decreased by a rate r per interval of time,
compounded n times per interval, its value A after t intervals of time is:
A 5 A0 A 1 1 nr B
nt
If the increase or decrease is continuous for t units of time, the formula
becomes
A 5 A0ert
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Review Exercises
315
VOCABULARY
7-4 Exponential growth • Exponential decay • Exponential function •
Horizontal asymptote • Natural base • Natural exponential function
7-6 Exponential equation
REVIEW EXERCISES
In 1–12, evaluate each expression.
1. 30
1
2. 221
5. 8(2)3
6. 5(0.1)22
1
7. 12 (9)
1
1
2
8. (2 3 3)22
12. A 12 3 43 B
2
10. 10022
9. (9 3 9) 4
4. 5(3)0
3. 83
11. 1253
232
In 13–16, evaluate each function for the given value. Be sure to show your work.
1
1
13. g(x) 5 x4x4; g(4)
15. g(x) 5 2x 1 A 21x B ; g(4)
14. g(x) 5 ¢ 1 1 ≤ ; g(3)
1 1 x23
16. g(x) 5 A xx7 B
5
22
; g(10)
In 17–20, write each expression with only positive exponents and express the
answer in simplest form. The variables are not equal to zero.
17. (27x) 3 ? 3(x) 21
19. A bx22 B 25
3
1
3bc 23
18. A aabc
3B
20. A 2x2y22z2 B (3x23y3z21)
3
In 21–24, write each radical expression as a power with positive exponents and
express the answer in simplest form. The variables are positive numbers.
3
21. !25x
23. "64a7b20
12
81y3
3
22. Ä
3y5
6
24. "32x8y3
In 25–28, write each power as a radical expression in simplest form. The variables are positive numbers.
5
1
25. (2y) 2
26. (16x8y) 2
27. Q
12 9
1
28. A a 6b 3c B 4
(a 1 2) 34
24 R
ab
29. a. Sketch the graph of y 5 1.25x.
b. On the same set of axes, sketch the graph of y 5 0.80x.
c. Under what transformation is the graph of y 5 0.80x the image of
y 5 1.25x?
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Exponential Functions
In 30–41, solve and check each equation.
1
1
30. x22 5 36
31. a3 5 4
33. y23 1 6 5 14
2
34. a31 7 5 9
32. b22 5 23
35. 4x 1 2 5 10
36. 2(5)2x 5 50
37. 7x 5 72x22
38. 2x12 5 8x22
39. 0.102x 5 102x11
40. 3x11 5 27x
41. 2 5 0.5x
a
42. Find the interest that has accrued on an investment of $500 if interest of
4% per year is compounded quarterly for a year.
43. The label on a prescription bottle directs the patient to take the medicine
twice each day. The effective ingredient of the medicine decreases continuously at a rate of 25% per hour. If a dose of medicine containing 1 milligram of the effective ingredient is taken, how much is still present 12
hours later when the second dose is taken?
44. A sum of money that was invested at a fixed rate of interest doubled in
value in 15 years. The interest was compounded yearly. Find the rate of
interest to the nearest tenth of a percent.
45. A company records the value of a machine used for production at $25,000.
As the machine ages, its value depreciates, that is, decreases in value. If the
depreciation is estimated to be 20% of the value of the machine at the end
of each year, what is the expected value of the machine after 6 years?
46. Determine the common solution of the system of equations:
15y 5 27x
5y 5 3x
Exploration
Prove that for all a:
a
a22
a. 3a 221 3 a 5 23
3
1 3
CUMULATIVE REVIEW
b.
4a 1 1 1 4
a21
? 2
1 16
a15
2
5 14
CHAPTERS 1–7
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. The expression 2x3(4 2 3x) 2 5x(x 2 2) is equal to
(1) 6x4 2 8x3 1 5x2 2 10x
(3) 6x4 1 8x3 2 5x2 1 10x
(2) 26x4 1 8x3 2 5x2 1 10x
(4) 26x4 1 8x3 2 5x2 2 10x
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Cumulative Review
317
2. If the domain is the set of integers, the solution set of x2 2 3x 2 4 , 0 is
(1) {21, 0, 1, 2, 3, 4}
(3) { . . . , 23, 22, 21, 4, 5, 6, . . . }
(2) {0, 1, 2, 3}
(4) { . . . , 24, 23, 22, 5, 6, 7, . . . }
1
21
3. The fraction 31
is equal to
923
(1) 21
(2) 1
3
(3) 13
(4) 52
27
4. An equation whose roots are 23 and 5 is
(1) x2 1 2x 2 15 5 0
(3) x2 1 2x 1 15 5 0
2
(2) x 2 2x 2 15 5 0
(4) x2 2 2x 1 15 5 0
5. In simplest form, !12 1 !9 1 !27 is
(1) 5 !3 1 3
(2) 6 !3
(3) 4 !3
6. The expression !22A !218B 1 !225 is equal to
(1) 1
(2) 26 1 5i
(3) 6 1 5i
(4) !48
(4) 11i
7. The discriminant of a quadratic equation is 35. The roots are
(1) unequal rational numbers.
(3) equal rational numbers.
(2) unequal irrational numbers.
(4) imaginary numbers.
8. In the sequence 100, 10, 1, . . . , a20 is equal to
(1) 10220
(2) 10219
(3) 10218
(4) 10217
9. The factors of x3 2 4x2 2 x 1 4 are
(1) (x 2 4)(x 2 1)(x 1 1)
(2) (x 2 2)(x 2 1)(x 1 1)(x 1 2)
(3) (x 1 4)(x 2 1)(x 1 1)
(4) (4 2 x)(x 2 1)(x 1 1)
10. The solution set of the equation 6 2 !7 2 x 5 8 is
(1) {3}
(2) {23}
(3) {23, 3}
(4) [
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Write an expression for the nth term of the series 1 1 3 1 5 1 7 1 .
12. The roots of a quadratic equation are 2 and 52. Write the equation.
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Exponential Functions
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
!24
13. Express 31 1
in a + bi form.
2 !24
14. Find the solution set of the equation 1 1 27x11 5 82.
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. What is the solution set of the inequality x2 2 7x 2 12 , 0?
16. A circle with center at (2, 0) and radius 4 is intersected by a line whose
slope is 1 and whose y-intercept is 2.
a. Write the equation of the circle in center-radius form.
b. Write the equation of the circle in standard form.
c. Write the equation of the line.
d. Find the coordinates of the points at which the line intersects the circle.
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CHAPTER
LOGARITHMIC
FUNCTIONS
The heavenly bodies have always fascinated and
challenged humankind. Our earliest records contain
conclusions, some false and some true, that were
believed about the relationships among the sun, the
moon, Earth, and the other planets. As more accurate
instruments for studying the heavens became available
and more accurate measurements were possible, the
mathematical computations absorbed a great amount
of the astronomer’s time.
A basic principle of mathematical computation is
that it is easier to add than to multiply. John Napier
(1550–1617) developed a system of logarithms that
facilitated computation by using the principles of exponents to find a product by using addition. Henry Briggs
(1560–1630) developed Napier’s concept using base
10. Seldom has a new mathematical concept been
more quickly accepted.
8
CHAPTER
TABLE OF CONTENTS
8-1 Inverse of an Exponential
Function
8-2 Logarithmic Form of an
Exponential Equation
8-3 Logarithmic Relationships
8-4 Common Logarithms
8-5 Natural Logarithms
8-6 Exponential Equations
8-7 Logarithmic Equations
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
319
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Logarithmic Functions
8-1 INVERSE OF AN EXPONENTIAL FUNCTION
In Chapter 7, we showed that any positive real number can be the exponent of a power by drawing the graph of the exponential function y 5 bx for
0 , b , 1 or b . 1. Since y 5 bx is a one-to-one function, its reflection in the
line y 5 x is also a function. The function x 5 by is the inverse function of
y 5 bx.
y
y
y = bx
x
=
y
=
x
y = bx
y
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x = by
O
x
O
b.1
x = by
x
0,b,1
The equation of a function is usually solved for y in terms of x. To solve the
equation x = by for y, we need to introduce some new terminology. First we will
describe y in words:
x 5 by: “y is the exponent to the base b such that the power is x.”
A logarithm is an exponent. Therefore, we can write:
x 5 by: “y is the logarithm to the base b of the power x.”
The word logarithm is abbreviated as log. Look at the essential parts of this
sentence:
y 5 logb x: “y is the logarithm to the base b of x.”
The base b is written as a subscript to the word “log.”
x 5 by can be written as y 5 logb x.
For example, let b 5 2. Write pairs of values for x 5 2y and y 5 log2 x.
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Inverse of an Exponential Function
x 5 2y
1
2
52
–1
1 5 20
!2 5 2
1
2
y 5 log2 x
In Words
–1 is the logarithm to the base 2 of
1
2.
0 is the logarithm to the base 2 of 1.
1
2
is the logarithm to the base 2 of !2.
21 5
log 2 12
0 5 log2 1
1
2
5 log 2 !2
321
(x, y)
A 12, 21 B
(1, 0)
A !2, 12 B
2 5 21
1 is the logarithm to the base 2 of 2.
1 5 log2 2
2
2 is the logarithm to the base 2 of 4.
2 5 log2 4
(4, 2)
3
3 is the logarithm to the base 2 of 8.
3 5 log2 8
(8, 3)
452
852
(2, 1)
We say that y 5 logb x, with b a positive number not equal to 1, is a
logarithmic function.
EXAMPLE 1
Write the equation x 5 10y for y in terms of x.
Solution x 5 10y ← y is the exponent or logarithm to the base 10 of x.
y 5 log10 x
Graphs of Logarithmic Functions
From our study of exponential functions in Chapter 7, we know that when
b . 1 and when 0 , b , 1, y 5 bx is defined for all real values of x. Therefore,
the domain of y 5 bx is the set of real numbers. When b . 1, as the negative values of x get larger and larger in absolute value, the value of bx gets smaller but
is always positive. When 0 , b , 1, as the positive values of x get larger and
larger, the value of bx gets smaller but is always positive. Therefore, the range of
y 5 bx is the set of positive real numbers.
When we interchange x and y to form the inverse function x 5 by or
y 5 logb x:
The domain of y 5 logb x is the set of positive real numbers.
The range y 5 logb x is the set of real numbers.
The y-axis or the line x 5 0 is a vertical asymptote of y 5 logb x.
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Logarithmic Functions
EXAMPLE 2
a. Sketch the graph of f(x) 5 2x.
b. Write the equation of f21(x) and sketch its graph.
Solution a. Make a table of values for
f(x) 5 2x, plot the points, and
draw the curve.
x
2x
f(x)
22
222 5 212
221 5 12
1
4
1
2
0
20
1
1
21
2
2
22
4
3
23
8
21
y
f(x) 2x
1 (x)
f
log 2
x
x
O
b. Let f(x) 5 2x → y 5 2x.
To write f21(x), interchange x
and y.
x 5 2y is written as y 5 log2 x. Therefore, f21(x) 5 log2 x.
To draw the graph, interchange x and y in each ordered pair or reflect
the graph of f(x) over the line y 5 x. Ordered pairs of f21(x) include A 14, 22 B ,
A 12, 21 B , (1, 0), (2, 1), (4, 2), and (8, 3).
y
The function y 5 logb x represents
logarithmic growth. Quantities represented
by a logarithmic function grow very slowly.
6
For example, suppose that the time it
takes for a computer to run a program
could by modeled by the logarithmic function y 5 log10 x where x is the number of
instructions of the computer program and y
x
O
is the running time in milliseconds. The
Number of
graph on the right shows the running time
instructions
in the interval 1 , x , 1,000,000. Note that
the graph increases from 0 to 5 for the first 100,000 instructions but only
increases from 5 to 6 as the number of instructions increases from 100,000 to
1,000,000. (Each interval on the x-axis represents 100,000 instructions.)
Running time (ms)
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Inverse of an Exponential Function
323
Exercises
Writing About Mathematics
1. Peg said that (1, 0) is always a point on the graph of y 5 logb x. Do you agree with Peg?
Explain why or why not.
2. Sue said that if x 5 b2y for b . 1, then y 5 12 logb x. Do you agree with Sue? Explain why or
why not.
Developing Skills
In 3–10: a. For each f(x), write an equation for f21(x), the inverse function. b. Sketch the graph of
f(x) and of f21(x).
3. f(x) 5 3x
4. f(x) 5 5x
x
7. f(x) 5 A 12 B
8. f(x) 5 A !2B x
x
5. f(x) 5 A 32 B
9. f(x) 5 31x
x
6. f(x) 5 A 52 B
10. f(x) 5 22x
In 11–22, solve each equation for y in terms of x.
11. x 5 6y
12. x 5 10y
13. x 5 8y
14. x 5 (0.1)y
15. x 5 (0.2)y
16. x 5 42y
17. x 5 12–y
18. x 5 log2 y
19. x 5 log5 y
20. x 5 log10 y
21. x 5 log8 y
22. x 5 log0.1 y
Applying Skills
23. Genevieve decided to organize a group of volunteers to help at a soup kitchen. Every week
for the first three weeks, the number of volunteers tripled so that the number, f(x), after x
weeks is f(x) 5 3x.
a. Write the ordered pairs of the function f(x) 5 3x for 0 # x # 3 and locate the pairs as
points on a graph. The domain is the set of non-negative integers.
b. Write the ordered pairs for f–1(x) and sketch the graph.
24. If money is invested at a rate of 5% compounded annually, then for each dollar invested,
the amount of money in an account is g(x), when g(x) 5 1.05x after x years.
a. Write the ordered pairs of the function g for 0 # x # 3 and locate the pairs as points
on a graph. The domain is the set of non-negative integers.
b. Write the ordered pairs for g–1(x) and sketch the graph.
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8-2 LOGARITHMIC FORM OF AN EXPONENTIAL EQUATION
An exponential equation and a logarithmic equation are two different ways of
expressing the same relationship. For example, to change an exponential equation to a logarithmic equation, recall that a logarithm or log is an exponent. In
the exponential equation 81 5 34, the exponent or log is 4. The basic statement
is:
log 5 4
Then write the base as a subscript of the word “log” to indicate the log to
the base 3:
log3 5 4
Now add the value of the power, that is, log to the base 3 of the power 81:
log3 81 5 4
To change from a logarithmic equation to an exponential statement, we
must again look for the exponent. For example, in the logarithmic equation
log10 0.01 5 22, the basic statement is that the log or exponent is 22. The base
is the number that is written as a subscript of the word “log”:
1022
The number that follows the word “log” is the power:
1022 5 0.01
1
5 0.01.
Recall that 1022 5 101 2 5 100
In general:
ba 5 c is equivalent to logb c 5 a.
Base
↓ ↓
ba 5 c
Power
↓
Exponent
Power
↓ ↓
logb c 5 a
Base
Exponent
↓
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EXAMPLE 1
Write 93 5 729 in logarithmic form and express its meaning in words.
Solution In the equation 93 5 729, the logarithm (exponent) is 3:
log 5 3
Write the base, 9, as a subscript of “log”:
log9 5 3
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Logarithmic Form of an Exponential Equation
325
Write the power, 729, after “log”:
log9 729 5 3
The logarithmic form of the equation means, “the exponent to the base 9 of the
power 729 is 3.” Answer
EXAMPLE 2
Write 23 5 log25 125 in exponential form.
Solution The equation 32 5 log25 125 says that 32 is the exponent. The base is written as
3
the subscript of “log.” The power, 252, is 125.
3
252 5 125 Answer
Note that 252 5 A252 B 3 5 A !25B 3 5 53 5 125.
3
1
EXAMPLE 3
Find the value of a when log8 a 5 31.
Solution The equation log8 a 5 13 means that for the power a, the base is 8, and the
exponent is 13.
1
83 5 a
3
"
85a
2 5 a Answer
EXAMPLE 4
Solve the following equation for x: log3 19 5 log2 x.
Solution Represent each side of the equation by y.
Let y 5 log3 19.
Then:
Then:
3y 5 91
y
22 5 log2 x
22
3 53
y 5 22
Answer x 5 14
Let y 5 log2 x.
222 5 x
1
4
5x
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Logarithmic Functions
Exercises
Writing About Mathematics
1. If logb c = a, explain why logb 1c 5 2a.
2. If logb c = a, explain why logb c2 5 2a.
Developing Skills
In 3–14, write each exponential equation in logarithmic form.
3. 24 5 16
4. 53 5 125
5. 64 5 82
6. 120 5 1
7. 216 5 63
8. 1021 5 0.1
9. 523 5 0.008
1
12. 643 5 4
11. 721 5 17
10. 422 5 0.0625
3
3
14. 0.001 5 10022
13. 6254 5 125
In 15–26, write each logarithmic equation in exponential form.
15. log10 100 5 2
16. log5 125 5 3
17. log4 16 5 2
18. 7 5 log2 128
19. 5 5 log3 243
20. log7 1 5 0
21. log10 0.001 5 23
22. log100 0.01 5 21
23. 22 5 log5 0.04
24. log8 2 5 31
25. log49 343 5 23
26. 225 5 log32 0.25
In 27–56, evaluate each logarithmic expression. Show all work.
27. log8 8
28. 5 log8 8
29. log6 216
30. 4 log6 216
31. log12 2
32. 8 log12 2
1
33. log4 16
1
34. 3 log4 16
35. log3 729
36. 31 log3 729
1
37. log4 64
1
38. 16 log4 64
39. log3 81
40. log2 16
41. log3 81 ? log2 16
42. log5 125
43. log10 1,000
44. log2 32
45. log12 14
46. 12 log3 81
47. log18 324
48. log6 36
49. log13 27
50. log1 627
3
51. log5 125 ? log10 1,000 ? 2 log2 32
6 log 36
52. log12 14 ? log3 81 ? 12 log18 324
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Logarithmic Relationships
53.
log5 25 1 2 log10 10
log16 4
1.5
54. log 64 2
log80 80
4
55.
3 log3 9 ? 4 log8 8 ? log13 169
6 log2 256 1 log12 8
56.
2 log
327
2.25
log3 27 1 8 log16 2
log8 512
? log1,000 10
In 57–68, solve each equation for the variable.
57. log10 x 5 3
58. log2 32 5 x
59. log5 625 5 x
60. a 5 log4 16
61. logb 27 5 3
62. logb 64 5 6
63. log5 y 5 22
64. log25 c 5 24
65. log100 x 5 212
66. log8 x 5 12
67. log36 6 5 x
68. logb 1,000 5 32
69. If f(x) 5 log3 x, find f(81).
70. If p(x) 5 log25 x, find p(5).
71. If g(x) 5 log10 x, find g(0.001).
72. If h(x) 5 log32 x, find h(8).
73. Solve for a: log5 0.2 5 loga 10
74. Solve for x: log100 10 5 log16 x
Applying Skills
75. When $1 is invested at 6% interest, its value, A, after t years is A 5 1.06t. Express t in terms
of A.
76. R is the ratio of the population of a town n years from now to the population now. If the
population has been decreasing by 3% each year, R 5 0.97n. Express n in terms of R.
77. The decay constant of radium is 20.0004/year. The amount of radium, A, present after t
years in a sample that originally contained 1 gram of radium is A 5 e–0.0004t.
a. Express 20.0004t in terms of A and e.
b. Solve for t in terms of A.
8-3 LOGARITHMIC RELATIONSHIPS
Because a logarithm is an exponent, the rules for exponents can be used to
derive the rules for logarithms. Just as the rules for exponents only apply to
powers with like bases, the rules for logarithms will apply to logarithms with the
same base.
Basic Properties of Logarithms
If 0 , b , 1 or b . 1:
b0 5 1 n logb 1 5 0
b1 5 b n logb b 5 1
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Logarithmic Functions
For example:
30 5 1 ↔ log3 1 5 1
31 5 3 ↔ log3 3 5 1
Logarithms of Products
If 0 , b , 1 or b . 1:
bx 5 c ↔ logb c 5 x
by 5 d ↔ logb d 5 y
bx1y 5 cd ↔ logb cd 5 (x + y)
Therefore,
logb cd 5 logb c 1 logb d
The log of a product is the sum of the logs of the factors of the product.
For example:
32 5 9 ↔ log3 9 5 2
34 5 81 ↔ log3 81 5 4
32+4 5 9 3 81 ↔ log3 (9 3 81) 5 log3 9 1 log3 81 5 2 1 4 5 6
EXAMPLE 1
If log5 125 5 3 and log5 25 5 2, find log5 (125 3 25).
logb cd 5 logb c 1 logb d
Solution
log5 (125 3 25) 5 log5 125 1 log5 25
log5 (125 3 25) 5 3 1 2
log5 (125 3 25) 5 5 Answer
Logarithms of Quotients
If 0 , b , 1 or b . 1:
bx 5 c ↔ logb c 5 x
by 5 d ↔ logb d 5 y
bx2y 5 dc ↔ logb dc 5 (x 2 y)
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Logarithmic Relationships
329
Therefore,
logb dc 5 logb c 2 logb d
The log of a quotient is the log of the dividend minus the log of the divisor.
For example:
32 5 9 ↔ log3 9 5 2
34 5 81 ↔ log3 81 5 4
9
9
↔ log3 A 81
3224 5 81
B 5 log3 9 2 log3 81 5 2 2 4 5 22
Do we get the same answer if the quotient is simplified?
9
81
5 19
log3 1 5 0
log3 A 19 B
log3 9 5 2
5 log3 1 2 log3 9 5 0 2 2 5 22 ✔
This is true because 3224 5 322 5 19.
EXAMPLE 2
Use logs to show that for b . 0, b0 5 1.
Solution Let logb c 5 a. Then,
logb c 2 logb c 5 logb A cc B
a 2 a 5 logb 1
0 5 logb 1
b0 5 1
Logarithms of Powers
If 0 , b , 1 or b . 1:
bx 5 c ↔ logb c 5 x
(bx)a 5 bax 5 ca ↔ logb ca 5 ax
Therefore,
logb ca = a logb c
The log of a power is the exponent times the log of the base.
For example:
32 5 9 ↔ log3 9 5 2
(32)3 5 3233 5 93 ↔ log3 (9)3 5 3 log3 9 5 3(2) 5 6
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Logarithmic Functions
EXAMPLE 3
Find log2 8: a. using the product rule, and b. using the power rule.
Solution We know that log2 2 5 1.
a. Product rule: log2 8 5 log2 (2 3 2 3 2) 5 log2 2 1 log2 2 1 log2 2
511111
53
b. Power rule: log2 8 5 log2 (23) 5 3 log2 2
5 3(1)
53
Check 23 5 8 ↔ log2 8 5 3
Answer Using both the product rule and the power rule, log2 8 5 3.
EXAMPLE 4
Find the value of log12 9 1 log12 16.
Solution Use the rule for product of logs:
log12 9 1 log12 16 5 log12 (9 3 16) 5 log12 144
Since 122 5 144, log12 144 5 2.
Answer log12 9 1 log12 16 5 2
EXAMPLE 5
in terms of x and y.
If log10 3 5 x and log10 5 5 y, express log10 !3
52
1
1
32
2
2
log10 !3
2 5 log10 52 5 log10 3 2 log10 5
5
Solution
5 12 log10 3 2 2 log10 5
5 12x 2 2y
Answer log10
!3
52
5 12x 2 2y
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Logarithmic Relationships
331
SUMMARY
Logarithms
Multiplication
logb cd 5 logb c 1 logb d
Division
logb dc 5 logb c 2 logb d
Logarithm of a Power
logb ca 5 a logb c
Logarithm of 1
logb 1 5 0
Logarithm of the Base
logb b 5 1
Exercises
Writing About Mathematics
1. Show that for all a . 0 and a 1, loga an 5 n.
2. Terence said that (loga b) ? (loga c) 5 loga bc. Do you agree with Terence? Explain why or
why not.
Developing Skills
In 3–14, use the table to the right and the properties of logarithms to evaluate each expression. Show
all work.
3. 27 3 81
4. 243 3 27
log3 19 5 22
log3 243 5 5
5. 19,683 4 729
6. 6,561 4 27
log3 13 5 21
log3 729 5 6
log3 1 5 0
log3 2,187 5 7
log3 3 5 1
log3 6,561 5 8
log3 9 5 2
log3 19,683 5 9
log3 27 5 3
log3 59,049 5 10
7. 9
4
8. 243
2
9. 81 3 9
4
11. !
243 3 2,187
13. 813 4 !729
2
10. !6,561 4 729
12. #19,683
2,187
3 729
14. 27 3 #
19,683
log3 81 5 4
In 15–20: a. Write each expression as a single logarithm. b. Find the value of each expression.
15. log3 1 1 log3 9
16. log3 27 1 log3 81
17. log3 243 2 log3 729
18. log3 6,561 2 log3 243
19. 31 log3 2,187 1 16 log3 81
20. log3 9 2 2 log3 27 1 log3 243
In 21–23: a. Expand each expression as a difference, sum, and/or multiple of logarithms. b. Find the
value of each expression.
9
21. 4 log3 27
22. 12 log3 3(243)
23. log4 "162
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Logarithmic Functions
In 24–29, write each expression as a single logarithm.
24. loge x 1 loge 10
25. log2 a 1 log2 b
26. 4 log2 (x 1 2)
27. log10 y 2 2 log10 (y 2 1)
28. loge x 1 2 loge y 2 2 loge z
29. 12 log3 x10 2 25 log3 x5
In 30–35, expand each expression using the properties of logarithms.
30. log2 2ab
31. log3 10
x
33. log10 (x 1 1) 2
34. log4 xy5
32. log5 a25
35. loge !x
6
In 36–47, write each expression in terms of A and B if log2 x 5 A and log2 y 5 B.
36. log2 xy
37. log2 x2y
38. log2 (xy)3
39. log2 xy3
40. log2 (x 4 y)
41. log2 (x2 4 y3)
42. log2 !xy
44. log2 !x
y3
43. log2 x !y
46. log2 x !x
45. log2 #xy
4
47. log2 !
y
In 48–53, solve each equation for the variable.
48. log2 23 1 log2 22 5 log2 x
49. log2 16 1 log2 2 5 log2 x
50. log2 x 2 log2 8 5 log2 4
51. log5 x 1 log5 x 5 log5 625
52. logb 64 2 logb 16 5 log4 16
53. log2 8 1 log3 9 5 logb 100,000
8-4 COMMON LOGARITHMS
DEFINITION
A common logarithm is a logarithm to the base 10.
When the base is 10, the base need not be written as a subscript of the word
“log.”
log10 100 5 log 100
Before calculators, tables of common logarithms were used for computation. The slide rule, a mechanical device used for multiplication and division, was
a common tool for engineers and scientists. Although logarithms are no longer
needed for ordinary arithmetic computations, they make it possible for us to
solve equations, particularly those in which the variable is an exponent.
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Common Logarithms
333
The TI-831/841 graphing calculator has a LOG key that will return the
common logarithm of any number. For example, to find log10 8 or log 8, enter
the following sequence of keys.
ENTER:
DISPLAY:
LOG
8 ENTER
log(8
.903089987
Therefore, log10 8 0.903089987 or 100.903089987 8.
To show that this last statement is true, store the value of log 8 as x, and then
use 10x , the 2nd function of the LOG key.
ENTER:
DISPLAY:
STO佡
X,T,⍜,n
2nd
ENTER
10x
X,T,⍜,n
ENTER
Ans >X
.903089987
>
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io (X
8
In the equation 100.903089987 5 8, the number 0.903089987 is the logarithm of
8 and 8 is the antilogarithm of 0.903089987. The logarithm is the exponent and
the antilogarithm is the power. In general:
For logb x 5 y, x is the antilogarithm of y.
EXAMPLE 1
Find log 23.75.
Solution Use the LOG key of the calculator.
ENTER:
DISPLAY:
LOG
23.75 ENTER
log(23.75
1.375663614
Answer log 23.75 1.375663614
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Logarithmic Functions
EXAMPLE 2
If log x 5 2.87534, find x to the nearest tenth.
Solution Log x 5 2.87534 can be written as 102.87534 5 x.
10x
ENTER:
2nd
DISPLAY:
key to find the antilogarithm.
10x
2.87534 ENTER
>
Use the
io (2.87534
750.4815156
Answer To the nearest tenth, log 750.5 5 2.87534 or 102.87534 5 750.5.
EXAMPLE 3
3
3 0.781
Use logarithms and antilogarithms to find the value of 4.20 4.83
to the nearest tenth.
Solution First take the log of the expression.
3
3 0.781
5 3 log 4.20 1 log 0.781 2 log 4.83
log 4.20 4.83
ENTER:
DISPLAY:
3 LOG 4.2
ⴙ
)
LOG
.781
)
ⴚ
LOG
4.83 ENTER
3log(4.2)+log(.7
81)-log(4.83
1.078451774
Now store this value for x and find its antilogarithm.
ENTER:
DISPLAY:
STO佡
X,T,⍜,n
ENTER
2nd
Ans >X
1.078451774
>
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io (X
11.97986087
3
3 0.781
Answer To the nearest tenth, 4.20 4.83
5 12.0.
10x
X,T,⍜,n
ENTER
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Common Logarithms
335
Exercises
Writing About Mathematics
1. Explain why log 80 5 1 1 log 8.
2. Explain why log x is negative if 0 , x , 1.
Developing Skills
In 3–14, find the common logarithm of each number to the nearest hundredth.
3. 3.75
4. 8.56
5. 47.88
6. 56.2
7. 562
8. 5,620
9. 0.342
10. 0.0759
11. 1
12. 10
13. 100
14. 0.1
In 15–23, evaluate each logarithm to the nearest hundredth.
15. log 1,024
16. log 80
17. log 0.002
18. log 9 1 log 3
19. log 64 2 log 16
20. 200 log 52
3 log 4
21. log 5
log 100 2 12 log 36
log 6
22.
23. log 12 ? log 100 ? log 300
In 24–35, for each given logarithm, find the antilogarithm, x. Write the answer to four decimal
places.
24. log x 5 0.5787
25. log x 5 0.8297
26. log x 5 1.3826
27. log x 5 1.7790
28. log x 5 2.2030
29. log x 5 2.5619
30. log x 5 4.8200
31. log x 5 20.5373
32. log x 5 20.05729
33. log x 5 21.1544
34. log x 5 23
35. log x 5 24
In 36–47, if log 3 5 x and log 5 5 y, write each of the logs in terms of x and y.
36. log 15
37. log 9
38. log 25
39. log 45
1
3
40. log 75
41. log 27
42. log
44. log 0.04
9
45. log 10
46. log 35
43. log 15
47. log A 35 B
2
In 48–55, if log a 5 c, express each of the following in terms of c.
48. log a2
49. log 10a
52. log 100
a
a
53. log 10
50. log 100 a
a
51. log 10
a
54. log A 10
55. log !a
B
2
56. Write the following expression as a single logarithm: log (x 2 4) 1 2 log 8 2 log 6.
2
2
57. Write the following expression as a multiple, sum, and/or difference of logarithms: log # z .
xy
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Applying Skills
logK
58. The formula t 5 0.045 log e gives the time t (in years) that it will take an investment P that is
compounded continuously at a rate of 4.5% to increase to an amount K times the original
principal.
a. Use the formula to complete the table to three decimal places.
K
1
2
3
4
5
10
20
30
t
log K
b. Use the table to graph the function t 5 0.045 log e.
c. If Paul invests $1,000 in a savings account that is compounded continuously at a rate of
4.5%, when will his investment double? triple?
59. The pH (hydrogen potential) measures the acidity or alkalinity of a solution. In general,
acids have pH values less than 7, while alkaline solutions (bases) have pH values greater
than 7. Pure water is considered neutral with a pH of 7. The pH of a solution is given by the
formula pH 5 2log x where x represents the hydronium ion concentration of the solution.
Find, to the nearest hundredth, the approximate pH of each of the following:
a. Blood: x 5 3.98 3 1028
b. Vinegar: x 5 6.4 3 1023
c. A solution with x 5 4.0 3 1025
8-5 NATURAL LOGARITHMS
DEFINITION
A natural logarithm is a logarithm to the base e.
Recall that e is an irrational constant that is approximately equal to
2.718281828. If x 5 ey, then y 5 loge x. The expression loge x is written as ln x.
Therefore, y 5 ln x can be read as “y is the natural log of x” or as “y is the exponent to the base e of x” and is called the natural logarithmic function.
The TI-831/841 graphing calculator has a LN key that will return the
natural logarithm of any number. For example, to find loge 8 or ln 8, enter the
following sequence of keys.
ENTER:
DISPLAY:
LN 8 ENTER
ln(8
2.079441542
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Natural Logarithms
337
Therefore, ln 8 2.079441542 or e2.079441542 8.
To show that this last statement is true, store the answer in your calculator
and then use the ex key, the 2nd function of the LN key.
ENTER:
DISPLAY:
STO佡
X,T,⍜,n
2nd
ENTER
ex
X,T,⍜,n
Ans >X
>
2.079441542
e (X
8
EXAMPLE 1
Find ln 23.75.
Solution Use the LN key of the calculator.
ENTER:
DISPLAY:
LN 23.75 ENTER
ln(23.75
3.16758253
Answer 3.16758253
EXAMPLE 2
If ln x 5 2.87534, find x to the nearest tenth.
Solution Use the ex key to find the antilogarithm.
ENTER:
DISPLAY:
2nd
ex
2.87534 ENTER
>
14411C08.pgs
e (2.87534
17.73145179
Answer To the nearest tenth, ln 17.7 5 2.87534 or e2.87534 5 17.7.
ENTER
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Logarithmic Functions
EXAMPLE 3
3
3 0.781
Use natural logs to find the value of 4.20 4.83
to the nearest tenth.
3
3 0.781
ln 4.20 4.83
5 3 ln 4.20 1 ln 0.781 2 ln 4.83
Solution
ENTER: 3 LN 4.2
DISPLAY:
ⴙ
)
LN .781
ⴚ
)
LN 4.83 ENTER
3ln(4.2)+ln(.781
)-ln(4.83
2.483226979
Now store this value for x and find its antilogarithm.
ENTER:
DISPLAY:
STO佡
X,T,⍜,n
ENTER
2nd
ex
X,T,⍜,n
ENTER
Ans >X
2.483226979
>
14411C08.pgs
e (X
11.97986087
3
3 0.781
Answer To the nearest tenth, 4.20 4.83
5 12.0.
Exercises
Writing About Mathematics
1. Compare Example 3 in Section 8-4 with Example 3 in Section 8-5. Explain why the answers
are the same.
2. For what value of a does log a 5 ln a? Justify your answer.
Developing Skills
In 3–14, find the natural logarithm of each number to the nearest hundredth.
3. 3.75
4. 8.56
5. 47.88
6. 56.2
7. 562
8. 5,620
9. 0.342
10. 0.0759
11. 1
12. e
13. e
14. 1e
2
In 15–20, evaluate each logarithm to the nearest hundredth.
15. ln 12
16. ln 5 1 ln 7
18. ln 1,0002
2 ln e
19. ln 62 ln
8
17. 12 ln 3 2 12 ln 1
20. lnln !5
10
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Natural Logarithms
In 21–32, for each given logarithm, find x, the antilogarithm. Write the answer to four decimal places.
21. ln x 5 0.5787
22. ln x 5 0.8297
23. ln x 5 1.3826
24. ln x 5 1.7790
25. ln x 5 2.2030
26. ln x 5 2.5619
27. ln x 5 4.8200
28. ln x 5 20.5373
29. ln x 5 20.05729
30. ln x 5 21.1544
31. ln x 5 21
32. ln x 5 22
In 33–44, if ln 2 5 x and ln 3 5 y, write each of the natural logs in terms of x and y.
33. ln 6
34. ln 9
35. ln 4
36. ln 12
37. ln 24
38. ln 36
39. ln 13
40. ln 12
41. ln 16
1
42. ln 36
43. ln 23
44. ln A 23 B
2
In 45–52, if ln a 5 c, express each of the following in terms of c.
48. ln a1
45. ln a2
46. ln a3
47. ln a21
49. ln a22
50. ln 12
51. ln a2
52. ln !a
55. ex 5 2
56. ex 5 22
1
a
In 53–56, find each value of x to the nearest thousandth.
53. ex 5 35
54. ex 5 217
57. Write the following expression as a single logarithm: 12 ln x 2 ln y 1 ln z3.
1
58. Write the following expression as a multiple, sum, and/or difference of logarithms: ln
e2xy2
z .
Hands-On Activity:The Change of Base Formula
You may have noticed that your calculator can only evaluate common and natural logarithms. For
logarithms with other bases, derive a formula by using the following steps:
1. Let x 5 logb y.
2. If x 5 logb y, then bx 5 y and log bx 5 log y. Use the rule for the logarithm of a power to
rewrite the left side of this last equation and solve for x.
3. Write logb y by equating the value of x from step 2 and the value of x from step 1. This
equation expresses the logarithm to the base b of y in terms of the common logarithm of y
and the common logarithm of b.
4. Repeat steps 2 and 3 using ln bx 5 ln y.
The equation in step 4 expresses the logarithm to the base b of y in terms of the natural logarithm of y and the natural logarithm of b.
These steps lead to the following equations:
log y
logb y 5 log b
ln y
logb y 5 ln b
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Logarithmic Functions
In a–h, use the calculator to evaluate the following logarithms to the nearest hundredth
using both common and natural logarithms.
a. log5 30
b. log2 17
c. log4 5
d. log3.5 10
e. log9 0.2
f. log12 6
g. log4 (12 3 80)
h. log5 15
28
8-6 EXPONENTIAL EQUATIONS
In chapter 7, we solved exponential equations by writing each side of the equation to the same base. Often that is possible only by using logarithms.
Since f(x) 5 logb x is a function:
If x1 5 x2, then logb x1 5 logb x2.
For example, solve 8 x 5 32 for x. There are two possible methods.
METHOD 1
Write each side of the equation
to the base 2.
METHOD 2
Take the log of each side of the equation
and solve for the variable.
8x 5 32
8x 5 32
(23)x 5 25
log 8x 5 log 32
3x 5 5
x log 8 5 log 32
log 32
x 5 53
x 5 log 8
ENTER:
LOG
32
8 ENTER
)
MATH
ⴜ
LOG
ENTER
ENTER
DISPLAY:
log(32)/log(8
1.666666667
AnsFrac
5/3
3 5
8B 5 25 5 32 ✔
Check: 83 5 A !
5
For exponential equations with different bases, we use logarithms. For
example, solve 5x 5 32 for x. In this equation, there is no base of which both 5
and 32 are a power.
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Exponential Equations
5x 5 32
Check
x
log 5 5 log 32
STO佡
ENTER:
x log 5 5 log 32
x5
ENTER:
LOG
LOG
DISPLAY:
32
5 ^
log 32
log 5
)
341
DISPLAY:
X,T,⍜,n
ENTER
X,T,⍜,n
ENTER
Ans >X
ⴜ
2.15338279
>
14411C08.pgs
5 X
5 ENTER
32
log(32)/log(5
2.15338279
When solving an exponential equation for a variable, either common logs or
natural logs can be used.
EXAMPLE 1
Solve for x to the nearest hundredth: 5.00(7.00)x 5 1,650.
Solution
How to Proceed
(1) Write the equation:
5.00(7.00)x 5 1,650
(2) Write the natural
log of each side of
the equation:
ln 5.00(7.00)x 5 ln 1,650
(3) Simplify the equation:
ln 5.00 1 ln 7.00x 5 ln 1,650
ln 5.00 1 x ln 7.00 5 ln 1,650
(4) Solve the equation for x:
x ln 7 5 ln 1,650 2 ln 5.00
x5
(5) Use a calculator to
compute x:
ENTER:
(
ln 1,650 2 ln 5.00
ln 7.00
LN 1650
LN 5
)
)
ⴚ
)
ⴜ
LN
7 ENTER
DISPLAY:
Answer x 2.98
(ln(1650)-ln(5))
/ln(7
2.980144102
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EXAMPLE 2
If a $100 investment receives 5% interest each year, after how many years will
the investment have doubled in value?
Solution Let A dollars represent the value of an investment of P dollars after t years at
interest rate r.
Since interest is compounded yearly, A = P(1 1 r)t.
If the value of the investment doubles, then A 5 2P.
P 5 100, A 5 2P 5 200, and (1 1 r) 5 (1 1 0.05) 5 (1.05).
Solve for t:
200 5 100(1.05)t
2 5 (1.05)t
log 2 5 t log 1.05
log 2
log (1.05)
ENTER:
DISPLAY:
LOG
2
)
ⴜ
5t
1.05
LOG
)
ENTER
log(2)/log(1.05)
14.20669908
The investment will have almost doubled after 14 years and will have doubled in value in the 15th year.
Answer 15 years
EXAMPLE 3
The element fermium has a decay constant of 20.00866/day. After how many
days will 7.0 grams remain of a 10.0-gram sample?
Solution Since an element decays constantly, use the equation An 5 A0ert.
A0 5 10.0, An 5 7.0, r 5 20.00866.
7.0 5 10.0 e20.00866t
0.70 5 e20.00866t
ln 0.70 5 ln e20.00866t
ln 0.70 5 20.00866t
ln 0.70
20.00866
5t
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Exponential Equations
ENTER:
DISPLAY:
LN .7
)
ⴜ
(-)
343
.00866 ENTER
ln(.7)/-.00866
41.18648313
The sample will decay to 7.0 grams on the 41st day.
Answer 41 days
Exercises
Writing About Mathematics
1. Trisha said that the equation 5x 1 6 5 127 could be solved by writing the logarithmic equation x log 5 1 log 6 5 log 127. Do you agree with Trisha? Explain why or why not.
2. Melita said that the equation 4(3)x 5 72 could be solved by writing the logarithmic equation
x log 12 5 log 72. Do you agree with Melita? Explain why or why not.
Developing Skills
In 3–14, solve each equation for the variable. Express each answer to the nearest hundredth.
3. 3x 5 12
4. 2b 5 18
5. 5y 5 100
6. 10x 5 50
7. 12a 5 254
8. 6(3x) 5 532
9. 7(2b) 5 815
12. (5 3 7)a 5 0.585
10. 5(10y) 5 1,200
11. (2 3 8)x 5 0.25
13. 12 1 9x 5 122
14. 75 2 4b 5 20
Applying Skills
15. When Rita was five, she had $1 in her piggy bank. The next year she doubled the amount
that she had in her piggy bank to $2. She decided that each year she would double the
amount in her piggy bank. How old will Rita be when she has at least $1,000 in her
piggy bank?
16. An investment of $2,000 receives 5% interest annually. After how many years has the
investment increased to at least $2,500?
17. When interest is compounded quarterly (4 times a year) at an annual rate of 6%, the rate of
interest for each quarter is 0.06
4 , and the number of times that interest is added in t years is
4t. After how many years will an investment of $100 compounded quarterly at 6% annually
nt
be worth at least $450? (Use the formula An 5 A0 A 1 1 nr B .)
18. After how many years will $100 invested at an annual rate of 6% compounded continuously
be worth at least $450? (Use the formula An 5 A0ert.)
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19. The decay constant of francium is 20.0315/minute.
a. After how many minutes will 1.25 grams of francium remain of a 10.0-gram sample?
Assume the exponential decay occurs continuously.
b. What is the half-life of francium? (The half-life of an element is the length of time
needed for half of a sample to decay. For example, it is the length of time for a sample
of 10 grams to be reduced to 5 grams of the original element.)
20. The half-life of einsteinium is 276 days.
a. To five decimal places, what is the decay constant of einsteinium? Assume the exponential decay occurs continuously.
b. After how many days will 2.5 grams of einsteinium remain of a sample of 20 grams?
8-7 LOGARITHMIC EQUATIONS
We have solved equations involving exponents by equating the log of each side
of the equation. An equation given in terms of logs can be solved by equating
the antilog of each side of the equation.
Since y 5 logb x is a one-to-one function:
If logb x1 5 logb x2, then x1 5 x2.
EXAMPLE 1
Solve for x and check: ln 12 2 ln x 5 ln 3.
Solution
How to Proceed
(1) Write the equation:
ln 12 2 ln x 5 ln 3
(2) Solve for ln x:
2ln x 5 2ln 12 1 ln 3
ln x 5 ln 12 2 ln 3
(3) Simplify the right side of the equation:
(4) Equate the antilog of each side of
the equation:
How to Proceed
Alternative
Solution (1) Write the equation:
(2) Simplify the left side of the equation:
ln x 5 ln 12
3
x 5 12
3
x54
ln 12 2 ln x 5 ln 3
ln 12
x 5 ln 3
(3) Equate the antilog of each side of the equation:
12
x
(4) Solve for x:
12 5 3x
53
45x
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Logarithmic Equations
Check
345
Calculator check
ENTER:
ln 12 2 ln x 5 ln 3
LN 12
ⴚ
)
LN 4 ENTER
?
ln 12 2 ln 4 5 ln 3
ln
12 ?
4 5
LN 3 ENTER
ln 3
DISPLAY:
ln 3 5 ln 3 ✔
ln(12)-ln(4
1.098612289
ln(3
1.098612289
Answer x 5 4
Note that in both methods of solution, each side must be written as a single
log before taking the antilog of each side.
EXAMPLE 2
Solve for x: log x 1 log (x 1 5) 5 log 6.
Solution (1) Write the equation:
log x 1 log (x 1 5) 5 log 6
(2) Simplify the left side:
log [x(x 1 5)] 5 log 6
(3) Equate the antilog of each side
of the equation:
x(x 1 5) 5 6
x2 1 5x 5 6
(4) Solve the equation for x:
x2 1 5x 2 6 5 0
(x 2 1)(x 1 6) 5 0
Reject the negative root. In the
given equation, log x is only
defined for positive values of x.
Check
x2150
x51
log x 1 log (x 1 5) 5 log 6
?
log 1 1 log (1 1 5) 5 log 6
?
0 1 log 6 5 log 6
log 6 5 log 6 ✔
Answer x 5 1
x1650
x 5 26 ✘
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EXAMPLE 3
Solve for b: logb 8 5 log4 64.
Solution Let each side of the equation equal x.
Let x 5 log4 64.
Let x 5 logb 8.
x
4 5 64
2 x
3 5 logb 8
6
b3 5 8
(2 ) 5 2
1
3
b5 !
8
2x 5 6
x53
Check
1
(b3) 3 5 83
22x 5 26
b52
logb 8 5 log4 64
?
log2 8 5 log4 64
353✔
Answer b 5 2
Exercises
Writing About Mathematics
1. Randall said that the equation log x 1 log 12 5 log 9 can be solved by writing the equation
x 1 12 5 9. Do you agree with Randall? Explain why or why not.
2. Pritha said that before an equation such as log x 5 1 1 log 5 can be solved, 1 could be
written as log 10. Do you agree with Pritha? Explain why or why not.
Developing Skills
In 3–14, solve each equation for the variable and check.
3. log x 1 log 8 5 log 200
4. log x 1 log 15 5 log 90
5. ln x 1 ln 18 5 ln 27
6. log x 2 log 5 5 log 6
7. log x 2 log 3 5 log 42
8. ln x 2 ln 24 5 ln 8
9. log 8 2 log x 5 log 2
10. log (x 1 3) 5 log (x 2 5) 1 log 3
11. log x 1 log (x 1 7) 5 log 30
12. log x 1 log (x 2 1) 5 log 12
13. 2 log x 5 log 25
14. 3 ln x 1 ln 24 5 ln 3
In 15–18, find x to the nearest hundredth.
15. log x 2 2 5 log 5
16. log x 1 log (x 1 2) 5 log 3
17. 2 log x 5 log (x 2 1) 1 log 5
18. 2 log x 5 log (x 1 3) 1 log 2
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Vocabulary
347
CHAPTER SUMMARY
A logarithm is an exponent: x = by ↔ y 5 logb x. The expression y 5 logb x
can be read as “y is the logarithm to the base b of x.”
For b . 0 and b 1, f(x) 5 bx, then f21(x) 5 logb x, the logarithmic function.
The domain of f21(x) 5 logb x is the set of positive real numbers and the range is
the set of real numbers. The y-axis is the vertical asymptote of f21(x) 5 logb x.
Because a logarithm is an exponent, the rules for logarithms are derived
from the rules for exponents. If bx 5 c and by 5 d:
Powers
Logarithms
b0 5 1
logb 1 5 0
Exponent of Zero
1
Exponent of One
b 5b
Products
xy
b
5 cd
logb cd 5 logb c 1 logb d
5x1y
Quotients
bxy 5 dc
logb dc 5 logb c 2 logb d
5x2y
(bx)a 5 bax 5 ca
logb ca 5 a logb c
5 ax
Powers
logb b 5 1
A common logarithm is a logarithm to the base 10. When no base is written,
the logarithm is a common logarithm: log A 5 log10 A.
For logb x 5 y, x is the antilogarithm of y.
A natural logarithm is a logarithm to the base e. A natural logarithm is
abbreviated ln: ln A 5 loge A.
The rules for logarithms are used to simplify exponential and logarithmic
equations:
• If logb x1 5 logb x2, then x1 5 x2.
• If x1 5 x2, then logb x1 5 logb x2.
VOCABULARY
8-1 Logarithm • Logarithm function • Vertical asymptote
8-4 Common logarithm • Antilogarithm
8-5 Natural logarithm • ln x • Natural logarithmic function
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REVIEW EXERCISES
1. a. Sketch the graph of f(x) 5 log3 x.
b. What is the domain of f(x)?
c. What is the range of f(x)?
d. Sketch the graph of f1(x).
e. Write an equation for f1(x).
In 2–4, solve each equation for y in terms of x.
2. x 5 log6 y
4. x 5 82y
3. x 5 log2.5 y
In 5–10, write each expression in logarithmic form.
5. 23 5 8
6. 62 5 36
8. 32 5 !3
9. 4 5 83
1
7. 101 5 0.1
2
10. 14 5 22
In 11–16, write each expression in exponential form.
11. log3 81 5 4
14. log7 !7 5 12
12. log5 125 5 3
13. log4 8 5 32
15. 21 5 log 0.1
16. 0 5 ln 1
In 17–20, evaluate each logarithmic expression. Show all work.
18. 14
9 log5 625
17. 3 log2 8
1
4 log9 !3
19. log3 27
20.
1
log2 256 ? log861 8614
1
4 log13 169
21. If f(x) 5 log3 x, find f(27).
22. If f(x) 5 log x, find f(0.01).
23. If f(x) 5 log4 x, find f(32).
24. If f(x) 5 ln x, find f(e4).
In 25–32, if log 5 5 a and log 3 5 b, express each log in terms of a and b.
25. log 15
27. log 5(3)2
26. log 25
3 5
29. log !3
30. log A 35 B
31. log #
3
5
33. If b . 1, then what is the value of logb b?
2
34. If log a 5 0.5, what is log (100a)?
28. log !45
32. log 25 39 !3
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Review Exercises
349
In 35–40, solve each equation for the variable. Show all work.
35. x 5 log4 32
36. logb 27 5 23
37. log6 x 5 22
38. log 0.001 5 x
39. log25 x 5 14
40. logb 16 5 22
In 41–44: a. Write each expression as a single logarithm. b. Find the value of each
expression.
41. 14 log4 81 2 log4 48
42. log360 5 1 log360 12 1 log360 6
43. log0.5 64 1 log0.5 0.25 2 8 log0.5 2
44. log1.5 23 1 log1.5 3 1 log1.5 21
In 45–47: a. Expand each expression using the properties of logarithms.
b. Evaluate each expression to the nearest hundredth.
2
45. ln 423
46. ln (142 ? 0.625)
47. ln (0.254 4 (26 ? 35))
In 48–53, write an equation for A in terms of x and y.
48. log A 5 log x 1 log y
49. log2 A 5 log2 x 2 log2 y
50. ln x 5 ln y 2 ln A
51. log5 A 5 log5 x 1 3 log5 y
52. log A 5 2(log x 2 log y)
53. ln x 5 ln A 213 ln y
In 54–56, if log x 5 1.5, find the value of each logarithm.
54. log 100x
55. log x2
56. log !x
57. If log x 5 log 3 2 log (x 2 2), find the value of x.
58. For what value of a does 1 1 log3 a 2 log3 2 5 log3 (a 1 1)?
59. If 2 log (x 1 1) 5 log 5, find x to the nearest hundredth.
60. In 2000, it was estimated that there were 50 rabbits in the local park and
that the number of rabbits was growing at a rate of 8% per year. If the
estimates are correct and the rabbit population continues to increase at
the estimated rate, in what year will there be three times the number of
rabbits, that is, 150 rabbits? (If A0 5 the number of rabbits present in 2000,
An 5 the number of rabbits n years later and k 5 the annual rate of
increase, An 5 A0ekn.)
61. When Tobey was born, his parents invested $2,000 in a fund that paid an
annual interest of 6%. How old will Tobey be when the investment is
worth at least $5,000?
62. The population of a small town is decreasing at the rate of 2% per year.
The town historian records the population at the end of each year. In 2000
(n 5 0), the population was 5,300. If this decrease continued, what would
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have been the population, to the nearest hundred, in 2008 (n 5 8)?
(Use Pn 5 P0(1 1 r)n where P0 is the population when n 5 0, Pn is the
population in n years, and r is the rate of change.)
Exploration
Using tables of logarithms was the common method of calculation before computers and calculators became everyday tools. These tables were relatively simple to use but their development was a complex process. John Napier spent
many years developing these tables, often referred to as Napier’s bones. Napier
observed a relationship between arithmetic and geometric sequences. To distinguish the arithmetic sequence from the geometric sequence, we will let an be the
terms of the arithmetic sequence and gn be the terms of the geometric sequence.
For example, compare the arithmetic sequence with a1 5 0 and d 5 1 with
the geometric sequence with g1 5 1 and r 5 10.
an
0
1
2
3
4
5
...
gn
1
10
100
1,000
10,000
100,000
...
Note that for all n, gn 5 10an or log gn 5 an.
The arithmetic mean between a1 and a2, that is, between 0 and 1, is
0 1 1
5 0.5. The geometric mean between g1 and g2 is the mean proportional
2
between the first and second terms of the sequence:
1
x
x
→ x2 5 (1)(10) or x 5 !(1)(10) 3.16227766
5 10
For these two means, 3.16227766 5 100.5 or log 3.16227766 5 0.5.
Now use a recursive method. Use the results of the last step to find a new log.
The arithmetic mean of 0 and 0.5 is 0 12 0.5 5 0.25. The geometric mean between
1 and 3.16227766 is 1.77827941. For these two means, 1.77827941 5 100.25 or
log 1.77827941 5 0.25.
Complete the following steps. Use a calculator to find the geometric means.
an
gn
0
1
1
10
(7)
(8)
(4)
(5)
(1)
(2)
2
100
STEP 1.
STEP 2.
STEP 3.
STEP 4.
STEP 5.
Find the arithmetic mean of a2 5 1 and a3 5 2 and place it in slot (1)
in the table on the left.
Find the geometric mean of g2 5 10 and g3 5 100 and place it in slot
(2) in the table on the left.
Show that the log of the geometric mean found in step 2 is equal to
the arithmetic mean found in step 1.
Repeat step 1 using a2 5 1 and the mean in slot (1) and place it in
slot (4).
Repeat step 2 using g2 5 10 and the mean in slot (2) and place it in
slot (5).
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Cumulative Review
STEP 6.
STEP 7.
STEP 8.
STEP 9.
STEP 10.
351
Show that the log of the geometric mean found in step 5 is equal to
the arithmetic mean found in step 4.
Repeat step 1 using a2 5 1 and the mean in slot (4) and place it in
slot (7).
Repeat step 2 using g2 5 10 and the mean in slot (5) and place it in
slot (8).
Show that the log of the geometric mean found in step 8 is equal to
the arithmetic mean found in step 7.
The geometric mean of 17.7827941 and 31.6227766 is 23.71373706.
Find log 23.71373706 without using the LOG or LN keys of the
calculator.
CUMULATIVE REVIEW
CHAPTERS 1–8
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. If the range of f(x) is a subset of the set of real numbers, which of the following is not an element of the domain of f(x) 5 "9 2 x2?
(1) 0
(2) 1
(3) 3
(4) 4
2. An equation with imaginary roots is
(1) x2 2 2x 2 5 5 0
(2) x2 2 2x 1 5 5 0
(3) x2 1 2x 2 5 5 0
(4) x2 1 2x 5 5
3. The solution set of 5 2 !x 1 1 5 x is
(1) {3, 8}
(2) {28, 23}
(3) {8}
(4) {3}
4. If f(x) 5 2x 1 3 and g(x) 5 x2, then f(g(22)) is equal to
(1) 24
(2) 1
(3) 4
(4) 11
5. Which of the following is an arithmetic sequence?
(1) 1, 3, 9, 27, 81
(3) 1, 3, 6, 12, 24
(2) 1, 2, 4, 7, 11
(4) 1, 4, 7, 10, 13
6. The expression (3 2 i)2 is equivalent to
(1) 8
(2) 8 2 6i
(3) 10
(4) 8 1 6i
7. The 9th term of the sequence 13, 9, 5, 1, . . . is
(1) 236
(2) 232
(3) 223
(4) 219
!3
8. The fraction 22 1
is equivalent to
2 !3
(1) 11 !3
(2) 7 24 !3
(4) 7 1 74 !3
(3) 7 1 4 !3
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Logarithmic Functions
9. The sum 2(3)2 1 61 is equal to
7
(1) 36
7
(2) 18
(3) 613
(4) 2(3)3
(2) 26
(3) 34
(4) 90
3
10. a 2ii is equal to
i50
(1) 24
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Find all the zeros of the function f(x) 5 4x3 2 x.
12. The sum of the roots of a quadratic equation is 10 and the product is 34.
Write the equation and find the roots.
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Find, to the nearest hundredth, the value of x such that 53x 5 1,000.
14. Express the sum of !200 1 !50 1 2 !8 in simplest radical form.
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
8
15. a. Write a 30(0.2) n21 as the sum of terms.
n51
8
b. Find a 30(0.2) n21 using a formula for the sum of a series.
n51
16. Carbon-14 has a decay constant of 20.000124 when time is in years. A
sample of wood that is estimated to have originally contained 15 grams of
carbon-14 now contains 9.25 grams. Estimate, to the nearest hundred
years, the age of the wood.
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CHAPTER
TRIGONOMETRIC
FUNCTIONS
A triangle is the most basic shape in our study of
mathematics. The word trigonometry means the
measurement of triangles.The study of the sun, Earth,
and the other planets has been furthered by an understanding of the ratios of the sides of similar triangles.
Eratosthenes (276–194 B.C.) used similar right triangles
to estimate the circumference of Earth to about
25,000 miles. If we compare this to the best modern
estimate, 24,902 miles, we see that even though his
methods involved some inaccuracies, his final results
were remarkable.
Although in the history of mathematics, the applications of trigonometry are based on the right triangle, the scope of trigonometry is much greater than
that. Today, trigonometry is critical to fields ranging
from computer art to satellite communications.
9
CHAPTER
TABLE OF CONTENTS
9-1 Trigonometry of the Right
Triangle
9-2 Angles and Arcs as Rotations
9-3 The Unit Circle, Sine, and
Cosine
9-4 The Tangent Function
9-5 The Reciprocal Trigonometric
Functions
9-6 Function Values of Special
Angles
9-7 Function Values from the
Calculator
9-8 Reference Angles and
the Calculator
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
353
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9-1 TRIGONOMETRY OF THE RIGHT TRIANGLE
The right triangle is an important geometric figure that has many applications
in the world around us. As we begin our study of trigonometry, we will recall
what we already know about right triangles.
A right triangle has one right angle and two complementary acute angles.
The side opposite the right angle is the longest side of the triangle and is called
the hypotenuse. Each side opposite an acute angle is called a leg.
In the diagram, ABC is a right trianB
gle with C the right angle and AB the
hypotenuse. The legs are BC, which is oppoe
us
n
e
site A, and AC, which is opposite B. The
t
leg
po
hy
leg AC is said to be adjacent to A and the
leg BC is also said to be adjacent to B.
The acute angles, A and B, are compleA
leg
C
mentary.
When the corresponding angles
of two triangles are congruent, the
B
E
triangles are similar. In the diagram,
ABC and DEF are right triangles with right angles C and F.
If A D, then B E
because complements of congruent A
D
F
C
angles are congruent. Therefore,
the corresponding angles of ABC
and DEF are congruent and
ABC DEF.
Two triangles are similar if and only if the lengths of their corresponding
sides are proportional. For similar triangles ABC and DEF, we can write:
AC
AB
5 DF
5 DE
In any proportion, we can interchange the means of the proportion to write
a new proportion:
BC
EF
BC
BC
AB
EF
5 DE
5 DE
• Since EF
, we can write AB
.
AC
AC
AB
DF
5 DE
5 DE
• Since DF
, we can write AB
.
BC
AC
BC
EF
• Since EF 5 DF, we can write AC 5 DF.
The ratio of any two sides of one triangle is equal to the ratio of the corresponding sides of a similar triangle.
Let ABC be a right triangle with a right angle at C. Any other right triangle that has an acute angle congruent to A is similar to ABC. In ABC, BC
is the leg that is opposite A, AC is the leg that is adjacent to A, and AB is
the hypotenuse. We can write three ratios of sides of ABC. The value of each
of these ratios is a function of the measure of A, that is, a set of ordered pairs
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Trigonometry of the Right Triangle
355
whose first element is the measure of A and whose second element is one and
only one real number that is the value of the ratio.
B
BC
sine of A 5 AB
5
cosine of A 5 AC
AB 5
length of the leg opposite /A
length of the hypotenuse
length of the leg adjacent to /A
length of the hypotenuse
length of the leg opposite /A
BC
tangent of A 5 AC
5 length of the leg adjacent to /A
A
C
We name each of these functions by using the first three letters of the name
of the function just as we named the absolute value function by using the first
three letters of the word “absolute.” We also abbreviate the description of the
sides of the triangle used in each ratio.
opp
sin A 5 hyp
adj
opp
cos A 5 hyp
tan A 5 adj
Note: A word or sentence can be helpful in remembering these ratios. Use the
first letter of the symbols in each ratio to make up a memory aid such as
“SohCahToa,” which comes from:
Adj
Opp
Sin A 5 Hyp
Opp
Cos A 5 Hyp
Tan A 5 Adj
EXAMPLE 1
In right triangle PQR, R is a right angle, PQ 5 25, QR 5 24, and PR 5 7. Find
the exact value of each trigonometric ratio.
a. sin P
b. cos P
c. tan P
d. sin Q
e. cos Q
Solution The hypotenuse is PQ and PQ 5 25.
f. tan Q
P
The side opposite P and the side adjacent to
Q is QR and QR 5 24.
25
7
R
Q
24
The side opposite Q and the side adjacent to
P is PR and PR 5 7.
opp
7
b. cos P 5 hyp 5 25
opp
e. cos Q 5 hyp 5 24
25
a. sin P 5 hyp 5 24
25
7
d. sin Q 5 hyp 5 25
opp
adj
c. tan P 5 adj 5 24
7
adj
7
f. tan Q 5 adj 5 24
opp
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Trigonometric Functions
EXAMPLE 2
An altitude to a side of an equilateral triangle bisects that side forming two congruent right triangles. One of the acute angles of each right triangle is an angle
of the equilateral triangle and has a measure of 60°. Show that sin 60° 5 !3
2 ,
1
cos 60° 5 2, and tan 60° 5 !3.
C
Solution Let s be the length of the equal sides, AC and
CB, of equilateral triangle ABC and CM the
altitude to AB. The midpoint of AB is M and
AM 5 2s .
s
A
Use the Pythagorean Theorem
to find CM:
(AC)2 5 (AM)2 1 (CM)2
s2 5
4s2
4
2
A 2s B
1 (CM)2
2
2 s4 5 (CM)2
3s2
4
s"3
2
5 (CM)2
s
2
s
2
M
B
Use the ratio of sides to find sine, cosine,
and tangent:
The measure of A is 60, hyp 5 AC 5 s,
s
opp 5 CM 5 s !3
2 , and adj 5 AM 5 2.
s !3
!3
2
sin 60° 5 hyp 5 CM
AC 5 s 5 2
opp
adj
5 CM
s
s
1
2
cos 60° 5 hyp 5 AM
AC 5 s 5 2
opp
CM
5
tan 60° 5 adj 5 AM
s !3
2
s
2
5 !3
Note that the sine, cosine, and tangent values are independent of the value of s
and therefore independent of the lengths of the sides of the triangle.
Exercises
Writing About Mathematics
1. In any right triangle, the acute angles are complementary. What is the relationship between
the sine of the measure of an angle and the cosine of the measure of the complement of
that angle? Justify your answer.
2. Bebe said that if A is the measure of an acute angle of a right triangle, 0 , sin A , 1. Do
you agree with Bebe? Justify your answer.
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357
Developing Skills
In 3–10, the lengths of the sides of ABC are given. For each triangle, C is the right angle and
mA , mB. Find: a. sin A b. cos A c. tan A.
3. 6, 8, 10
7. 16, 30, 34
4. 5, 12, 13
8. 2 !5, 2, 4
5. 11, 60, 61
9. !2, 3, !7
6. 8, 17, 15
10. 6, 3 !5, 9
11. Two of the answers to Exercises 3–10 are the same. What is the relationship between the triangles described in these two exercises? Justify your answer.
12. Use an isosceles right triangle with legs of length 3 to find the exact values of sin 45°,
cos 45°, and tan 45°.
13. Use an equilateral triangle with sides of length 4 to find the exact values of sin 30°, cos 30°,
and tan 30°.
Applying Skills
14. A 20-foot ladder leaning against a vertical wall reaches to a height of 16 feet. Find the sine,
cosine, and tangent values of the angle that the ladder makes with the ground.
15. An access ramp reaches a doorway that is 2.5 feet above the ground. If the ramp is 10 feet
long, what is the sine of the angle that the ramp makes with the ground?
16. The bed of a truck is 5 feet above the ground. The driver of the truck uses a ramp 13 feet
long to load and unload the truck. Find the sine, cosine, and tangent values of the angle that
the ramp makes with the ground.
17. A 20-meter line is used to keep a weather balloon in place. The sine of the angle that the
line makes with the ground is 34. How high is the balloon in the air?
18. From a point on the ground that is 100 feet from the base of a building, the tangent of
the angle of elevation of the top of the building is 45. To the nearest foot, how tall is the
building?
19. From the top of a lighthouse 75 feet high, the cosine of the angle of depression of a boat out
at sea is 45. To the nearest foot, how far is the boat from the base of the lighthouse?
9-2 ANGLES AND ARCS AS ROTATIONS
Many machines operate by turning a wheel. As a wheel turns, each spoke from
the center of the wheel moves through a central angle and any point on the rim
of the wheel moves through an arc whose degree measure is equal to that of the
central angle. The angle and the arc can have any real number as their measure.
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Trigonometric Functions
For example, the diagram shows a rotation
of three right angles or 270° as point A moves
to point B. The rotation is counterclockwise
(opposite to the direction in which the hands of
a clock move). A counterclockwise rotation is
said to be a positive rotation. Therefore,
X
A
O
X
mAOB 5 mAXB 5 270.
When a wheel turns in a clockwise direction
(the same direction in which the hands of a
clock turn), the rotation is said to be negative.
For example, the diagram shows a rotation of
2200° as point C moves to point D. Therefore,
X
B
D
O
mCOD 5 mCXD 5 2200. The ray at which
C
an angle of rotation begins is the initial side
h
of the angle. Here, OC is the initial side of
COD. The ray at which an angle of rotation
ends is the terminal side of the angle. Here,
X
h
OD is the terminal side of COD.
In order to study rotations through angles and
arcs, and the lengths of line segments associated with
these angles and arcs, we compare the rotations with
central angles and arcs of a circle in the coordinate
plane. An angle is in standard position when its vertex
is at the origin and its initial side is the nonnegative
ray of the x-axis.
We classify angles in standard position according
to the quadrant in which the terminal side lies. The
Greek symbol u (theta) is often used to represent
angle measure.
y
terminal
side
O
• If 0 , u , 90 and mAOB 5 u 1 360n
for any integer n, then AOB is a
first-quadrant angle.
x
y
O
• If 90 , u , 180 and mAOB 5 u 1 360n
for any integer n, then AOB is a
second-quadrant angle.
initial
side
B
B
u 1 360n
A x
y
u 1 360n
O
A x
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Angles and Arcs as Rotations
• If 180 , u , 270 and mAOB 5 u 1 360n
for any integer n, then AOB is a
third-quadrant angle.
359
y
u 1 360n
A x
O
B
• If 270 , u , 360 and mAOB 5 u 1 360n
for any integer n, then AOB is a
fourth-quadrant angle.
y
O
A x
u 1 360n
B
An angle in standard position whose terminal side lies on either the x-axis
or the y-axis is called a quadrantal angle.
The degree measure of a quadrantal angle is a multiple of 90.
The following figures show examples of quadrantal angles:
y
90°
O
x
y
y
180°
270°
O
x
O
y
O
x
290°
x
Coterminal Angles
When D on the positive ray of the x-axis is rotated
counterclockwise about the origin 270° to point D9, a
quadrantal angle in standard position, DOD9, is
formed. When point D on the positive ray of the x-axis
›
›
is rotated clockwise 90° to point D0, ODr and ODs
coincide. We say that DOD0 and DOD9 are coterminal angles because they have the same terminal
side. The measure of DOD9 is 270° and the measure
of DOD0 is 290°. The measure of DOD0 and the
measure of DOD9 differ by 360° or a complete rotation. The measure of DOD0, 290, can be written as
270 1 360(21) or mDOD9 plus a multiple of 360.
y
270°
D
O
90°
D D
x
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DEFINITION
Angles in standard position that have the same terminal side are coterminal
angles.
If AOB and AOC are angles in standard position with the same terminal side, then mAOB 5 mAOC 1 360n for some integer n.
EXAMPLE 1
An angle in standard form with a measure of 500° lies in what quadrant?
Solution Find a coterminal angle of the 500° angle with a
measure between 0° and 360°.
The measures of coterminal angles can be found by
adding or subtracting multiples of 360°, the measure
of a complete rotation.
y
140°
500°
x
500 2 360 5 140
Therefore, angles of 500° and 140° are coterminal.
An angle of 140° is in quadrant II.
An angle of 500° is a second-quadrant angle. Answer
EXAMPLE 2
Find the measures of five angles that are coterminal with an angle of 120°.
Solution If angles are coterminal, then their degree measures differ by a multiple of
360o. The measures of angles coterminal with an angle of 120o are of the form
120 1 360n for any integer n. Five of the infinitely many coterminal angle
measures are given below:
120 1 360(1) 5 480
120 1 360(2) 5 840
120 1 360(21) 5 2240
120 1 360(22) 5 2600
120 1 360(3) 5 1,200
Angles whose measures are 2600°, 2240°, 480°, 840°, and 1,200° are all
coterminal with an angle of 120°. Answer
Exercises
Writing About Mathematics
1. Is an angle of 810° a quadrantal angle? Explain why or why not.
2. Huey said that if the sum of the measures of two angles in standard position is a multiple of
360, then the angles are coterminal. Do you agree with Huey? Explain why or why not.
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Angles and Arcs as Rotations
Developing Skills
In 3–7, draw each angle in standard position.
3. 45°
4. 540°
5. 2180°
6. 2120°
7. 110°
In 8–17, name the quadrant in which an angle of each given measure lies.
8. 25°
13. 2200°
9. 150°
14. 2280°
10. 200°
11. 300°
12. 275°
15. 2400°
16. 750°
17. 1,050°
In 18–27, for each given angle, find a coterminal angle with a measure of u such that 0 u , 360.
18. 390°
19. 412°
20. 1,000°
21. 210°
22. 285°
23. –270°
24. 2500°
25. 540°
26. 360°
27. 980°
Applying Skills
28. Do the wheels of a car move in the clockwise or counterclockwise direction when the car is
moving to the right of a person standing at the side of the car?
29. To remove the lid of a jar, should the lid be turned clockwise or counterclockwise?
30. a. To insert a screw, should the screw be turned clockwise or counterclockwise?
b. The thread spirals six and half times around a certain screw. How many degrees should
the screw be turned to insert it completely?
31. The blades of a windmill make one complete rotation per second. How many rotations do
they make in one minute?
32. An airplane propeller rotates 750 times per minute. How many times will a point on the
edge of the propeller rotate in 1 second?
33. The Ferris wheel at the county fair takes 2 minutes to complete one full rotation.
a. To the nearest second, how long does it take the wheel to rotate through an angle
of 260°?
b. How many minutes will it take for the wheel to rotate through an angle of 1,125°?
30(360°) 5 10,800° per minute
Find the angular speed in degrees per second of a tire rotating:
a. 3 times per minute
b. 90 times per minute
c. 600 times per minute
angu
la
P
peed
rs
34. The measure of angle POA changes as P is rotated around the origin. The ratio of the change in the measure of the angle to the time
it takes for the measure to change is called the angular speed of
point P. For example, if a ceiling fan rotates 30 times per minute, its
angular speed in degrees per minute is:
O
A
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9-3 THE UNIT CIRCLE, SINE, AND COSINE
Wheels, machinery, and other items that rotate suggest that trigonometry must
apply to angles that have measures greater than a right angle and that have negative as well as positive measures. We want to write definitions of sine, cosine,
and tangent that include the definitions that we already know but go beyond
them. We will use a circle in the coordinate plane to do this.
A circle with center at the origin and radius 1 is called the unit circle and
has the equation x2 1 y2 5 1. Let u be the measure of a central angle in standard
position.
Let AOB be an angle in standard posiy
tion with its terminal side in the first quadB
rant and mAOB 5 u. Let P(p, q) be the
P(p, q)
1
point at which the terminal side intersects
u q
the unit circle and Q be the point at which
p
O
Q R(1, 0) A x
the vertical line through P intersects the
x-axis. Triangle POQ is a right triangle with
right angle at Q, mQOP 5 mAOB 5 u,
PQ 5 q, OQ 5 p, and OP 5 1.
adj
OQ
p
opp
PQ
q
cos u 5 hyp 5 OP 5 1 5 p
sin u 5 hyp 5 OP 5 1 5 q
Therefore, the coordinates of P are (cos u, sin u). We have shown that for a
first-quadrant angle in standard position, the cosine of the angle measure is the
x-coordinate of the point at which the terminal side of the angle intersects the
unit circle and the sine of the angle measure is the y-coordinate of the same
point. Since P is in the first quadrant, p and q are positive numbers and cos u
and sin u are positive numbers.
We can use the relationship between the point P on the unit circle and angle
POR in standard position to define the sine and cosine functions for any angle,
not just first-quadrant angles. In particular, notice that POR with measure u
determines the y-coordinate of P for any value of u. This y-value is defined to
be sin u. Similarly, POR with measure u determines the x-coordinate of P for
any value of u. This x-value is defined to be cos u.
Let us consider angle POR in each quadrant.
CASE 1
First-quadrant angles
POR is a first-quadrant angle.
The coordinates of P are (p, q).
cos u 5 p is positive.
sin u 5 q is positive.
y
u
O
P(p, q)
x
R(1, 0)
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The Unit Circle, Sine, and Cosine
CASE 2
Second-quadrant angles
y
P(p, q)
POR is a second-quadrant angle.
The coordinates of P are (p, q).
cos u 5 p is negative.
sin u 5 q is positive.
CASE 3
u
O
Third-quadrant angles
y
POR is a third-quadrant angle.
The coordinates of P are (p, q).
cos u 5 p is negative.
sin u 5 q is negative.
CASE 4
u
O
P(p, q)
Fourth-quadrant angles
P
R
u
O
R(1, 0)
x
P(p, q)
Quadrantal angles
y
O
x
R(1, 0)
y
POR is a fourth-quadrant angle.
The coordinates of P are (p, q).
cos u 5 p is positive.
sin u 5 q is negative.
CASE 5
x
R(1, 0)
y
P
u
O
x
R
y
u
P
O
x
x
u
R
O
R
P
u 5 0°
cos u 5 1
sin u 5 0
u 5 90°
cos u 5 0
sin u 5 1
u 5 180°
cos u 5 21
sin u 5 0
u 5 270°
cos u 5 0
sin u 5 21
These examples allow us to use the unit circle to define the sine and cosine
functions for the measure of any angle or rotation.
DEFINITION
Let POQ be an angle in standard position and P
be the point where the terminal side of the angle
intersects the unit circle. Let mPOQ 5 u. Then:
• The sine function is the set of ordered pairs x
(u, sin u) such that sin u is the y-coordinate of P.
• The cosine function is the set of ordered pairs
(u, cos u) such that cos u is the x-coordinate of P.
y
P
u
O
R(1, 0)
Q
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Trigonometric Functions
EXAMPLE 1
If P A 212, !3
2 B is a point on the unit circle and on the
y
P
terminal side of an angle in standard position whose
u
measure is u, find: a. sin u b. cos u
Solution a. sin u 5 y-coordinate of P 5
b. cos u 5 x-coordinate of P 5
!3
2 .
212.
x
O
R(1, 0)
Answer
Answer
EXAMPLE 2
y
Angle AOB is an angle in standard position with
measure u. If u 5 72° and A is a point on the unit
circle and on the terminal side of AOB, find the
coordinates of A to the nearest hundredth.
A
u
O
Solution The coordinates of A 5 (cos 72°, sin 72°).
x
B(1, 0)
Since cos 72° 0.3091 and sin 72° 0.9515, the
coordinates of A to the nearest hundredth are
(0.31, 0.95). Answer
EXAMPLE 3
Find sin (2450°) and cos (2450°).
Solution An angle in standard position whose measure is
2450 has the same terminal side as an angle of
2450 1 2(360) or 270°. Let P be the point where
the terminal side of an angle of 270° intersects the
unit circle. The coordinates of P are (0, 21).
sin (2450°) 5 sin 270° 5 21
y
270°
x
450°
P(0, 1)
cos (2450°) 5 cos (2450°) 5 0
Therefore, sin (2450°) 5 21 and cos (2450°) 5 0. Answer
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365
EXAMPLE 4
In the diagram OPT is an equilateral triangle
with O at the origin, P on the unit circle,
T(21, 0), and R(1, 0). Let u 5 mROP. Find:
a. u b. cos u c. sin u
Solution a. Since ROP and TOP are a linear pair
of angles, they are supplementary. The
measure of an angle of an equilateral
triangle is 60°. Therefore, mTOP 5 60
and mROP 5 120 or u 5 120.
y
P
u
T(1, 0)
R(1, 0)
x
QO
b. The value of cos u or cos 120 is equal to the x-coordinate of P. Draw PQ,
the altitude from P in OPT. In an equilateral triangle, the altitude bisects
the side to which it is drawn. Since OT 5 1, OQ 5 21. The x-coordinate of P
is negative and equal to 2OQ. Therefore, cos u 5 2OQ 5 212.
c. The value of sin u or sin 120 is equal to the y-coordinate of P. Triangle OQP
is a right triangle with OP 5 1.
OP 2 5 OQ2 1 PQ2
12 5 A 12 B 1 PQ2
1 5 14 1 PQ2
2
4
4
2 14 5 PQ2
3
4
!3
2
5 PQ2
5 PQ
The y-coordinate of P is positive and equal to PQ. Therefore, sin u 5 PQ 5 !3
2 .
Answers a. 120° b. cos 120° 5 221 c. sin 120° 5
!3
2
The values given here are exact values. Compare these values with the
rational approximations given on a calculator. Press MODE on your calculator. In the third line, DEGREE should be highlighted.
ENTER:
SIN
2nd
120
)
¯ 3
ENTER
)
ENTER:
ⴜ
COS
120
)
ENTER
21 ⴜ 2 ENTER
2 ENTER
DISPLAY:
sin(120)
.8660254038
√(3)/2
.8660254038
DISPLAY:
cos(120)
-.5
-1/2
-.5
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Trigonometric Functions
Exercises
Writing About Mathematics
1. If P is the point at which the terminal side of an angle in standard position intersects the
unit circle, what are the largest and smallest values of the coordinates of P? Justify your
answer.
2. Are the sine function and the cosine function one-to-one functions? Justify your answer.
Developing Skills
In 3–10, the terminal side of ROP in standard position intersects the unit circle at P. If mROP
is u, find: a. sin u b. cos u c. the quadrant of ROP
3. P A 35, 45 B
5
, 212
7. P A 213
13 B
4. P(0.6, 20.8)
7
8. P A 24
25, 225 B
1
5. P A 2 !3
2 , 2B
!2
9. P A !2
2 , 2 B
2 !5
6. P A !5
5 ,2 5 B
9 40
, 41 B
10. P A 241
In 11–14, for each of the following function values, find u if 0° u , 360°.
11. sin u 5 1,
cos u 5 0
12. sin u 5 21,
cos u 5 0
13. sin u 5 0,
cos u 5 1
14. sin u 5 0,
cos u 5 21
In 15–22, for each given angle in standard position, determine to the nearest tenth the coordinates
of the point where the terminal side intersects the unit circle.
15. 90°
16. 2180°
17. 81°
18. 137°
19. 229°
20. 312°
21. 540°
22. 245°
23. If P A 23, y B is a point on the unit circle and on the terminal side of an angle in standard position with measure u, find: a. y b. sin u c. cos u
Applying Skills
24. Let ROP be an angle in standard position, P the point at which the terminal side intersects the unit circle and mROP 5 u.
a. Under the dilation Da, the image of A(x, y) is A9(ax, ay). What are the coordinates P9
of the image of P under the dilation D5?
b. What are the coordinates of P0, the image of P under the dilation D25?
c. Express mROP9 and mROP0 in terms of u.
25. Under a reflection in the y-axis, the image of A(x, y) is A9(2x, y). The measure of
ROP 5 u and P(cos u, sin u) is a point on the terminal side of ROP. Let P9 be the
image of P and R9 be the image of R under a reflection in the y-axis.
a. What are the coordinates of P9?
b. Express the measure of R9OP9 in terms of u.
c. Express the measure of ROP9 in terms of u.
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The Unit Circle, Sine, and Cosine
Hands-On Activity
Use graph paper, a protractor, a ruler,
and a pencil, or a computer program for
this activity.
y
1. Let the side of each square of the grid
represent 0.1. Draw a circle with center
at the origin and radius 1.
P
2. Draw the terminal side of an angle of
20° in standard position. Label the
point where the terminal side intersects
the circle P.
0.1
O 0.1
x
3. Estimate the coordinates of P to the
nearest hundredth.
4. Use a calculator to find sin 20° and
cos 20°.
5. Compare the numbers obtained from
the calculator with the coordinates of P.
6. Repeat steps 2 through 5, replacing 20° with the following values: 70°, 100°, 165°, 200°, 250°,
300°, 345°.
In each case, are the values of the sine and the cosine approximately equal to the coordinates of P?
Hands-On Activity: Finding Sine and Cosine Using Any Point on the Plane
For any ROP in standard position, point P(x, y) is on the terminal
side, OP 5 r 5 "x2 1 y2, and mROP 5 u. The trigonometric function values of u are:
sin u 5
y
r
cos u 5
x
r
y
P(x, y)
u
O
Note that the point P is no longer restricted to the unit circle.
In (1)–(8), the given point is on the terminal side of angle u. Find r, sin u, and cos u.
(1) (3, 4)
(2) (25, 12)
(3) (8, 215)
(4) (22, 27)
(5) (1, 7)
(6) (21, 7)
(7) (1, 22)
(8) (23, 23)
R
x
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Trigonometric Functions
9-4 THE TANGENT FUNCTION
We have used the unit circle and an angle in standard position to write a general definition for the sine and cosine functions of an angle of any measure. We
can write a similar definition for the tangent function in terms of a line tangent
to the unit circle.
y
We will begin with an angle in the first
quadrant. Angle ROP is an angle in standard
P T(1, t)
position and P is the point at which the terminal
side of ROP intersects the unit circle. Let
x
u
mROP 5 u. At R, the point at which the unit
O
R(1, 0)
circle intersects the positive ray of the x-axis, construct a line tangent to the circle and let T be the
point at which the terminal side intersects this
tangent line. Recall that a line tangent to the circle is perpendicular to the radius drawn to the
point at which the tangent line intersects the circle. Since OR is a portion of
the x-axis, RT is a vertical line. The x-coordinate of T is the same as the
x-coordinate of R. The coordinates of T are (1, t). Since ORT is a right angle
and ORT is a right triangle, OR 5 1 and RT = t.
opp
RT
5 1t 5 t
tan u 5 adj 5 OR
Therefore, the coordinates of T are (1, tan u). We have shown that for a firstquadrant angle in standard position with measure u, tan u is equal to the y-coordinate of the point where the terminal side of the angle intersects the line that
is tangent to the unit circle at the point (1, 0). Since T is in the first quadrant, t
is a positive number and tan u is a positive number.
We can use this relationship between the point T and the tangent to the unit
circle to define the tangent function for angles other than first-quadrant angles.
In particular, ROP with measure u determines the y-coordinate of T for any
value of u. This y-value is defined to be tan u.
Let us consider angle ROP in each quadrant.
CASE 1
y
First-quadrant angles
P T(1, t)
x
ROP is a first-quadrant angle.
The coordinates of T are (1, t).
tan u 5 t is positive.
CASE 2
u
O
Second-quadrant angles
ROP is a second-quadrant angle.
Extend the terminal side through O.
The coordinates of T are (1, t).
tan u 5 t is negative.
R(1, 0)
y
P
u
O
R(1, 0)
x
T(1, t)
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The Tangent Function
CASE 3
Third-quadrant angles
y
ROP is a third-quadrant angle.
Extend the terminal side through O.
The coordinates of T are (1, t).
tan u 5 t is positive.
CASE 4
O
y
R(1, 0)
u
x
P T(1, t)
Quadrantal angles
y
y
P
u
x
R5P
5 T(1, 0)
u 5 0°
tan 0° 5 0
R(1, 0)
P
Fourth-quadrant angles
y
O
T(1, t)
x
u
ROP is a fourth-quadrant angle.
The coordinates of T are (1, t).
tan u 5 t is negative.
CASE 5
369
O
u
x
R(1, 0)
y
P
O
x
x
u
O
R 5 T(1, 0)
R(1, 0)
P
u 5 90°
no intersection
tan 90° is
undefined
u 5 180°
tan 180° 5 0
u 5 270°
no intersection
tan 270° is
undefined
We have extended the definition of the tangent function on the unit circle
for any angle or rotation for which tan u is defined and to determine those
angles or rotations for which tan u is not defined.
DEFINITION
y
Let T be the point where the terminal side of
an angle in standard position intersects the
line that is tangent to the unit circle at
R(1, 0). Let mROP 5 u. Then:
• The tangent function is a set of ordered
pairs (u, tan u) such that tan u is the
y-coordinate of T.
P T(1, t)
u
O
x
R(1, 0)
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Trigonometric Functions
EXAMPLE 1
Line l is tangent to the unit circle at A(1, 0). If
TA1, 2!5B is a point on l, what is tan AOT?
Solution tan AOT 5 y-coordinate of T
5 2!5 Answer
y
l
O
A
x
T
EXAMPLE 2
Angle AOT is an angle in standard position with measure u. If u 5 242° and T
is a point on the line tangent to the unit circle at A(1, 0), find the coordinates of
T to the nearest hundredth.
Solution The coordinates of T 5 (1, tan 242°).
Since tan 242° 1.8807, the coordinates of T to the nearest hundredth are
(1, 1.88). Answer
EXAMPLE 3
Angle AOB is an angle in standard position and mAOB 5 u. If sin u , 0 and
tan u , 0, in what quadrant is AOB?
Solution When the sine of an angle is negative, the angle could be in quadrants III or
IV. When the tangent of an angle is negative, the angle could be in quadrants
II or IV. Therefore, when both the sine and the tangent of the angle are
negative, the angle must be in quadrant IV. Answer
EXAMPLE 4
g
In the diagram, RT is tangent to the unit circle
at R(1, 0). The terminal ray of ROP intersects
the unit circle at P(p, q) and the line that contains the terminal ray of ROP intersects the
tangent line at T(1, t). The vertical line from P
intersects the x-axis at Q(p, 0).
sin u
a. Prove that if mROP 5 u, then tan u 5 cos
u.
1
b. If the coordinates of P are A 22 !2
3 , 23 B , find
sin u, cos u, and tan u.
y
Q
u
O
P(p, q)
T(1, t)
x
R(1, 0)
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The Tangent Function
371
Solution a. In the diagram, RT and PQ are vertical segments perpendicular to the
x-axis. Therefore, ROT and QOP are right triangles. The vertical angles
ROT and QOP are congruent. Therefore, ROT QOP by AA.
The ratios of corresponding sides of similar triangles are in proportion.
(1) Write a proportion using the lengths of the
legs of the right triangles. The lengths of
the legs are positive numbers. RT 5 t,
QP 5 q, and OQ 5 p:
(2) Since t is a positive number, t 5 t. Since p
and q are negative numbers, p 5 2p and
q 5 2q:
(3) Replace t, q, and p with the function values
that they represent:
QP
RT
OR
5 OQ
t
1
5 p
t
1
5 2p
q
2q
q
t5p
sin u
tan u 5 cos
u
2 !2
1
1
b. The coordinates of P are A 22 !2
3 , 23 B , cos u 5 2 3 , sin u 5 23, and:
sin u
tan u 5 cos
u 5
213
22!2
3
1
1
5 2 !2
5 2 !2
3 !2
5 !2
4 Answer
!2
What we have established in the last example is that for a given value of u,
the tangent function value is equal to the ratio of the sine function value to the
cosine function value.
sin u
tan u 5 cos
u
EXAMPLE 5
Use the properties of coterminal angles to find tan (2300°).
Solution Coterminal angles are angles in standard position
that have the same terminal side.
2300 1 360 5 60
2300°
Therefore, a 2300° angle is coterminal with a 60°
angle.
sin (2300°) 5 sin 60° 5 !3
2
cos (2300°) 5 cos 60° 5 12
!3
tan (2300°) 5 sin (23008) 5 21 5 !3 Answer
cos (23008)
y
2
O
60°
x
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Trigonometric Functions
Exercises
Writing About Mathematics
1. a. What are two possible measures of u if 0° , u , 360° and sin u 5 cos u? Justify your
answer.
b. What are two possible measures of u if 0° , u , 360° and tan u 5 1? Justify your
answer.
2. What is the value of cos u when tan u is undefined? Justify your answer.
Developing Skills
In 3–11, P is the point at which the terminal side of an angle in standard position intersects the unit circle. The measure of the angle is u. For each point P,
the x-coordinate and the quadrant is given. Find: a. the y-coordinate of P
b. cos u c. sin u d. tan u
3. A 35, y B , first quadrant
4.
6
, y B , second quadrant
5. A 210
6.
7. A 2 !3
2 , y B , second quadrant
8.
9. A 2 !2
2 , y B , third quadrant
10.
11. A 34, y B , first quadrant
12.
y
P
u
O
A 135 , y B , fourth quadrant
x
A 214, y B , third quadrant
A !5
3 , y B , first quadrant
A 15, y B , fourth quadrant
A 2 !7
3 , y B , second quadrant
In 13–20, P is a point on the terminal side of an angle in standard position with measure u and on a
circle with center at the origin and radius r. For each point P, find: a. r b. cos u c. sin u d. tan u
13. (7, 24)
14. (8, 15)
15. (21, 21)
16. (23, 24)
17. (23, 6)
18. (22, 6)
19. (4, 24)
20. (9, 23)
In 21–26, if u is the measure of AOB, an angle in standard position, name the quadrant in which
the terminal side of AOB lies.
21. sin u . 0, cos u . 0
22. sin u , 0, cos u . 0
23. sin u , 0, cos u , 0
24. sin u . 0, tan u . 0
25. tan u , 0, cos u , 0
26. sin u , 0, tan u . 0
27. When sin u 5 21, find a value of:
28. When tan u 5 0, find a value of:
a. cos u
a. sin u
29. When tan u is undefined, find a value of:
b. tan u
b. cos u
a. sin u
c. u
c. u
b. cos u
c. u
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The Tangent Function
Applying Skills
30. Angle ROP is an angle in standard position with
mROP 5 u, R(1, 0) a point on the initial side of ROP,
and P(p, q) the point at which the terminal side of ROP
intersects the unit circle.
y
P(p, q)
u
a. What is the domain of cos u and of sin u?
O
R(1, 0)
x
b. What is the range of cos u and of sin u?
c. Is tan u defined for all angle measures?
d. What is the domain of tan u?
e. What is the range of tan u?
31. Use the definitions of sin u and cos u based on the unit circle to prove that sin2 u 1 cos2 u 5 1.
32. Show that if ROP is an angle in standard position and mROP 5 u, then the slope of
g
PO 5 tan u.
Hands-On Activity
Use graph paper, protractor, ruler, and
pencil, or geometry software for this
activity.
1. On graph paper, let the side of each
square represent 0.1. Draw a circle with
center at the origin and radius 1 and the
line tangent to the circle at (1, 0).
2. Draw the terminal side of an angle of 20°.
Label point T, the point at which the terminal side intersects the tangent.
y
T
0.1
O 0.1
x
3. Estimate the y-coordinate of T to the
nearest hundredth.
4. Use a calculator to find tan 20.
5. Compare the number obtained from the
calculator with the y-coordinates of T.
6. Repeat steps 2 through 5, replacing 20° with the following values: 70°, 100°, 165°, 200°, 250°,
300°, 345°.
In each case, are the values of the tangent approximately equal to the y-coordinate of T?
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Trigonometric Functions
9-5 THE RECIPROCAL TRIGONOMETRIC FUNCTIONS
We have defined three trigonometric functions: sine, cosine, and tangent. There
are three other trigonometric functions that can be defined in terms of sin u,
cos u, and tan u. These functions are the reciprocal functions.
The Secant Function
DEFINITION
The secant function is the set of ordered pairs (u, sec u) such that for all u for
1
which cos u 0, sec u 5 cos
u.
We know that cos u 5 0 for u 5 90, u 5 270, and for any value of u that differs from 90 or 270 by 360. Since 270 5 90 1 180, cos u 5 0 for all values of u
such that u 5 90 1 180n where n is an integer. Therefore, sec u is defined for the
degree measures of angles, u, such that u 90 1 180n.
The secant function values are the reciprocals of the cosine function values.
We know that 21 cos u 1. The reciprocal of a positive number less than
or equal to 1 is a positive number greater than or equal to 1. For example, the
reciprocal of 43 is 43 and the reciprocal of 29 is 92. Also, the reciprocal of a negative
number greater than or equal to 21 is a negative number less than or equal to
21. For example, the reciprocal of 278 is 287 and the reciprocal of 215 is 25.
Therefore:
21 cos u , 0 → sec u 21
0 , cos u 1 → sec u 1
The set of secant function values is the set of real numbers that are
less than or equal to 21 or greater than or equal to 1, that is,
{x : x $ 1 or x # 21}.
The Cosecant Function
DEFINITION
The cosecant function is the set of ordered pairs (u, csc u) such that for all u
1
for which sin u 0, csc u 5 sin
u.
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The Reciprocal Trigonometric Functions
375
We know that sin u 5 0 for u 5 0, for u 5 180, and for any value of u that
differs from 0 or 180 by 360. Therefore, sin u 5 0 for all values of u such that
u 5 180n for all integral values of n. Therefore, csc u is defined for the degree
measures of angles, u, such that u 180n.
The cosecant function values are the reciprocals of the sine function values.
We know that 21 sin u 1. Therefore, the cosecant function values is the
same set of values as the secant function values. Therefore:
21 sin u , 0 → csc u 21
0 , sin u 1 → csc u 1
The set of cosecant function values is the set of real number that are
less than or equal to 21 or greater than or equal to 1, that is,
{x : x 1 or x 21}.
The Cotangent Function
DEFINITION
The cotangent function is the set of ordered pairs (u, cot u) such that for all u
1
for which tan u is defined and not equal to 0, cot u 5 tan
u and for all u for
which tan u is undefined, cot u 5 0.
We know that tan u 5 0 for u 5 0, for u 5 180, and for any value of u that
differs from 0 or 180 by 360. Therefore, tan u 5 0 for all values of u such that
u 5 180n, when n is an integer. Therefore, cot u is defined for the degree measures of angles, u, such that u 180n.
The cotangent function values are the real numbers that are 0 and the reciprocals of the nonzero tangent function values. We know that the set of tangent
function values is the set of all real numbers. The reciprocal of any nonzero real
number is a nonzero real number. Therefore, the set of cotangent function values is the set of real numbers. Therefore:
tan u , 0 → cot u , 0
tan u . 0 → cot u . 0
tan u is undefined → cot u 5 0
The set of cotangent function values is the set of real numbers.
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Trigonometric Functions
Function Values in the Right Triangle
In right triangle ABC, C is a right
angle, AB is the hypotenuse, BC is the
side opposite A, and AC is the side
adjacent to A. We can express the
three reciprocal function values in terms
of these sides.
B
hyp
opp
adj
C
opp
BC
sin A 5 hyp 5 AB
A
hyp
AB
1 5
csc A 5 sin1 A 5 opp
opp 5 BC
hyp
cos A 5
adj
hyp
tan A 5
opp
adj
5
AC
AB
5
BC
AC
sec A 5
1
cos A
cot A 5
1
tan A
1 5 hyp 5 AB
5 adj
AC
adj
hyp
1 5 adj 5 AC
5 opp
opp
BC
adj
EXAMPLE 1
The terminal side of an angle in standard position intersects the unit circle at
P Q 57, 22 !6
7 R . If the measure of the angle is u, find the six trigonometric function
values.
Solution If the coordinates of the point at which the terminal side of the angle intersects the unit circle are Q 57, 22 !6
7 R , then:
sin u 5 22 !6
7
cos u 5 57
22 !6
sin u
7
5
5 22 !6
tan u 5 cos
u
5
5
7
!6
7
7
csc u 5 22 !6
5 22 !6
3 !6
5 2712
!6
sec u 5 75
!6
5 !6
5
5
1
cot u 5 tan
u 5 22 !6 5 22 !6 3 !6 5 12
EXAMPLE 2
Show that (tan u)(csc u) 5 sec u.
Solution Write tan u and csc u in terms of sin u and cos u.
sin u
1
1
(tan u)(csc u) 5 cos
u A sin u B 5 cos u 5 sec u
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377
Exercises
Writing About Mathematics
1. Explain why sec u cannot equal 0.5.
sin u
2. When tan u is undefined, cot u is defined to be equal to 0. Use the fact that tan u 5 cos
u and
1
cot u 5 tan u to explain why it is reasonable to define cot 90 5 0.
Developing Skills
In 3–10, the terminal side of ROP in standard position intersects the unit circle at P. If mROP
is u, find: a. sin u b. cos u c. tan u d. sec u e. csc u f. cot u
3. P(0.6, 0.8)
4. P(0.96, 20.28)
1
7. P Q 2 !2
3 , 3R
2
8. P Q2 !5
3 , 23 R
5. P Q216, !35
6 R
3 !2
9. P Q2 !7
5 , 5 R
6. P Q212, !3
2 R
3
10. P Q22 !10
7 , 27 R
In 11–18, P is a point on the terminal side of an angle in standard position with measure u and on a
circle with center at the origin and radius r. For each point P, find: a. r b. csc u c. sec u d. cot u
11. (3, 4)
12. (1, 4)
13. (23, 23)
14. (25, 25)
15. (26, 6)
16. (24, 8)
17. (9, 29)
18. (9, 23)
19. If sin u 5 0, find all possible values of: a. cos u b. tan u c. sec u
20. If cos u 5 0, find all possible values of: a. sin u b. cot u c. csc u
21. If tan u is undefined, find all possible values of: a. cos u b. sin u c. cot u
22. If sec u is undefined, find all possible values of sin u.
23. What is the smallest positive value of u such that cos u 5 0?
Applying Skills
24. Grace walked within range of a cell phone tower. As soon as her cell phone received a signal, she looked up at the tower. The cotangent of the angle of elevation of the top of the
1
tower is 10
. If the top of the tower is 75 feet above the ground, to the nearest foot, how far is
she from the cell phone tower?
25. A pole perpendicular to the ground is braced by a wire 13 feet long that is fastened to the
ground 5 feet from the base of the pole. The measure of the angle the wire makes with the
ground is u. Find the value of:
a. sec u
b. csc u
c. cot u
26. An airplane travels at an altitude of 6 miles. At a point on the ground, the measure of the
angle of elevation to the airplane is u. Find the distance to the plane from the point on the
ground when:
a. csc u 5 2
b. sec u 5 53
c. cot 5 !3
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27. The equation sin2 u 1 cos2 u 5 1 is true for all u.
y
T(1, tan u)
P(cos u, sin u)
a. Use the given equation to prove that
tan2 u 1 1 5 sec2 u.
b. Is the equation in part a true for all u? Justify your
answer.
x
R(1, 0)
u
O
c. In the diagram, if OR 5 1 and RT 5 tan u, the length
of what line segment is equal to sec u?
u
28. Show that cot u 5 cos
sin u for all values of u for which
sin u 0.
9-6 FUNCTION VALUES OF SPECIAL ANGLES
Equilateral triangles and isosceles right triangles occur frequently in our study
of geometry and in the applications of geometry and trigonometry. The angles
associated with these triangles are multiples of 30 and 45 degrees. Although a
calculator will give rational approximations for the trigonometric function values of these angles, we often need to know their exact values.
The Isosceles Right Triangle
The measure of an acute angle of an isosceles right triangle is 45 degrees. If the measure of a leg is 1, then the
length of the hypotenuse can be found by using the
Pythagorean Theorem.
c2 5 a2 1 b2
AB2 5 12 1 12
2
AB 5 1 1 1
AB 5 !2
B
1
Therefore:
1
1
sin 45° 5 hyp 5 !2
5 !2
3 !2
5 !2
2
!2
opp
adj
1
1
5 !2
3 !2
5 !2
cos 45° 5 hyp 5 !2
2
!2
opp
1
tan 45° 5 adj 5 1 5 1
45°
1
C
A
The Equilateral Triangle
C
An equilateral triangle is separated into two congruent right triangles by an altitude from any vertex. In
the diagram, CD is altitude from C, ACD BCD,
CDA is a right angle, mA 5 60, and mACD 5 30.
If AC 5 2, AD 5 1.
2
A
60°
1 D
B
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Function Values of Special Angles
c2 5 a2 1 b2
2
2
Therefore:
2
2 5 1 1 CD
4 5 1 1 CD2
2
3 5 CD
!3 5 CD
379
sin 60° 5 hyp 5 !3
2
opp
opp
sin 30° 5 hyp 5 12
cos 30° 5 hyp 5 !3
2
adj
adj
cos 60° 5 hyp 5 12
1
1
tan 30° 5 adj 5 !3
tan 60° 5 adj 5 !3
5 !3
3 !3
5 !3
1 5 !3
3
!3
opp
opp
The 0°, 30°, 45°, 60°, and 90° angles and their multiples occur frequently and it
is useful to memorize the exact trigonometric function values summarized below:
u
0°
30°
sin u
0
cos u
1
tan u
0
1
2
!3
2
!3
3
45°
!2
2
!2
2
1
60°
90°
!3
2
1
1
2
0
!3
undefined
EXAMPLE 1
Use a counterexample to show that sin 2u 2 sin u.
Solution Let u 5 30° and 2u 5 3(30°) 5 60°.
1
sin 2u 5 sin 60° 5 !3
2 and 2 sin u 5 2 sin 30° 5 2 A 2 B 5 1.
Therefore, there exists at least one value of u for which sin 2u 2 sin u.
EXAMPLE 2
What are the lengths of the diagonals of a rhombus if the measure of a side is
10 centimeters and the measure of an angle is 120 degrees?
Solution The diagonals of a rhombus are perpendic- D
ular, bisect each other, and bisect the angles
of the rhombus. Therefore, the diagonals of
this rhombus separate it into four congruent
right triangles.
Let ABCD be the given rhombus with E
the midpoint of AC and BD, mDAB 5 120
and AB 5 10 centimeters. Therefore, AEB
is a right angle and mEAB 5 60.
cos 60 5 AE
AB
AE
1
2 5 10
2AE 5 10
AE 5 5
Answers AC 5 2AE 5 2(5) 5 10
BD 5 2BE 5 2A5 !3B 5 10 !3
C
E
60°
A
BE
sin 60 5 AB
!3
BE
2 5 10
2BE 5 10 !3
BE 5 5 !3
10 cm
B
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EXAMPLE 3
What is the exact value of sec 30°?
Solution sec 30° 5
1
cos 308
2
2
5 1 5 !3
5 !3
3 !3
5 2 !3
3 Answer
!3
!3
2
Exercises
Writing About Mathematics
1. R is the point (1, 0), P9 is the point on a circle with center at the origin, O, and radius r, and
mROP9 5 u. Alicia said that the coordinates of P9 are (r cos u, r sin u). Do you agree with
Alicia? Explain why or why not.
2. Hannah said that if cos u 5 a, then sin u 5 6"1 2 a2. Do you agree with Hannah? Explain
why or why not.
Developing Skills
In 3–44, find the exact value.
3. cos 30°
4. sin 30°
5. csc 30°
6. tan 30°
7. cot 30°
8. cos 60°
9. sec 60°
10. sin 60°
11. csc 60°
12. tan 60°
13. cot 60°
14. cos 45°
15. sec 45°
16. sin 45°
17. csc 45°
18. tan 45°
19. cot 45°
20. cos 180°
21. sec 180°
22. sin 180°
23. csc 180°
24. tan 180°
25. cot 180°
26. cos 270°
27. sec 270°
28. sin 270°
29. csc 270°
30. tan 270°
31. cot 270°
32. sin 450°
33. sin 0° 1 cos 0° 1 tan 0°
34. sin 45° 1 cos 60°
35. sin 90° 1 cos 0° 1 tan 45°
36. (cos 60°)2 1 (sin 60°)2
37. (sec 45°)2 2 (tan 45°)2
38. (sin 30°)(cos 60°)
39. (tan 45°)(cot 45°)
40. (sin 45°)(cos 45°)(tan 45°)
41. (sin 30°)(sec 60°)
308
42. tan
cos 608
sin 458
43. cos
458
sin 308
44. csc
308
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381
Applying Skills
45. A diagonal path across a rectangular field makes an angle of 30 degrees with the longer side
of the field. If the length of the path is 240 feet, find the exact dimensions of the field.
46. Use a counterexample to show that sin A 1 sin B 5 sin (A + B) is false.
47. Use a counterexample to show that A , B implies cos A , cos B is false.
9-7 FUNCTION VALUES FROM THE CALCULATOR
We used our knowledge of an equilateral triangle and of an isosceles right triangle to find the trigonometric function values for some special angles. How can
we find the trigonometric function values of any number?
Before calculators and computers were common tools, people who worked
with trigonometric functions used tables that supplied the function values. Now
these function values are stored in most calculators.
Compare the values given by a calculator to the exact values for angles of
30°, 45° and 60°. On your calculator, press MODE . The third line of that menu
lists RADIAN and DEGREE. These are the two common angle measures. In a
later chapter, we will work with radians. For now, DEGREE should be highlighted on your calculator.
We know that cos 30° 5 !3
2 . When you enter cos 30° into your calculator,
the display will show an approximate decimal value.
ENTER:
COS
DISPLAY:
cos(30)
30
)
ENTER
2nd
¯ 3
)
ⴜ
2 ENTER
.8660254038
√(3)/2
.8660254038
These two displays tell us that 0.8660254038 is a rational approximation of
the exact irrational value of cos 30°.
A degree, like an hour, is divided into 60 minutes. The measure of an angle
written as 37° 459 is read 37 degrees, 45 minutes, which is equivalent to 3745
60
degrees. Use the following calculator key sequence to find the tangent of an
angle with this measure.
ENTER:
DISPLAY:
TAN
37 ⴙ 45 ⴜ 60
tan(37+45/60)
.7742827278
)
ENTER
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Trigonometric Functions
EXAMPLE 1
Find sin 70° to four decimal places.
Solution Use a calculator.
ENTER:
DISPLAY:
SIN
70
)
ENTER
sin(70)
.9396926208
Round the decimal value. Since the digit following the fourth decimal place is
greater than 4, add 1 to the digit in the fourth decimal place.
Answer sin 70° 0.9397
EXAMPLE 2
Find sec 54° to four decimal places.
Solution sec 54° 5
ENTER: 1
DISPLAY:
1
o
cos 54
ⴜ
54
COS
)
ENTER
1/cos(54)
1.701301617
Round the decimal value. Since the digit following the fourth decimal place is
less than 5, drop that digit and all digits that follow.
Answer sec 54° 1.7013
EXAMPLE 3
Find sin 215° to four decimal places.
Solution ENTER:
DISPLAY:
SIN
215
)
ENTER
sin(215)
-.5735764364
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383
Recall that when the degree measure of an angle is greater than 180 and less
than 270, the angle is a third-quadrant angle. The y-coordinate of the intersection of the terminal side with the unit circle is negative. Therefore, the sine of
the angle is negative.
Answer 20.5736
Degree Measures of Angles
If sin u 5 0.9205, then u is the measure of an angle whose sine is 0.9205. In a circle, the degree measure of a central angle is equal to the degree measure of the
intercepted arc. Therefore, we can also say that u is the measure of an arc whose
sine is 0.9205. This can be written in symbols.
sin u 5 0.9205 → u 5 the angle whose sine is 0.9205
sin u 5 0.9205 → u 5 the arc whose sine is 0.9205
The words “the arc whose sine is” are abbreviated as arcsine. We further
shorten arcsine to arcsin.
sin u 5 0.9205 → u 5 arcsin 0.9205
On a calculator, the symbol for arcsin is “sin–1.”
sin u 5 0.9205 → u 5 sin–1 0.9205
We can also write:
cos u 5 0.3907 → u 5 arccos 0.3907 5 cos–1 0.3907
tan u 5 2.3559 → u 5 arctan 2.3559 5 tan–1 2.3559
Arccos and cos–1 are abbreviations for arccosine. Similarly, arctan and tan–1
are abbreviations for arctangent.
EXAMPLE 4
Find u to the nearest degree if cos u 5 0.8988 and 0° , u , 180°.
Solution cos u 5 0.8988 → u 5 arccos 0.8988 5 cos–1 0.8988
ENTER:
DISPLAY:
2nd
COSⴚ1 0.8988
)
ENTER
c o s - 1( 0 . 8 9 8 8 )
25.99922183
To the nearest degree, u 5 26°. Answer
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Trigonometric Functions
EXAMPLE 5
Find u to the nearest minute if tan u 5 3.5782 and 290° , u , 90°.
Solution The calculator will return the value of u as a whole number of degrees followed by a decimal value. To change the decimal part of the number to minutes, multiply by 60.
ENTER:
2nd
DISPLAY:
TANⴚ1 3.5782
)
ENTER
t a n - 1( 3 . 5 7 8 2 )
74.38590988
Now subtract 74 from this answer and multiply the difference by 60.
ENTER:
(
DISPLAY:
2nd
ANS
ⴚ
74
)
ⴛ
60 ENTER
t a n - 1( 3 . 5 7 8 2 )
74.38590988
(Ans-74)* 60
23.15459294
To the nearest minute, u 5 74° 239. Answer
Exercises
Writing About Mathematics
1. Explain why the calculator displays an error message when TAN 90 is entered.
2. Explain why the calculator displays the same value for sin 400° as for sin 40°.
Developing Skills
In 3–38, find each function value to four decimal places.
3. sin 28°
4. cos 35°
5. tan 78°
6. cos 100°
7. sin 170°
8. tan 200°
9. tan 20°
10. cos 255°
11. cos 75°
12. sin 280°
13. sin 80°
14. tan 375°
15. tan 15°
16. cos 485°
17. cos 125°
18. sin (210°)
19. sin 350°
20. sin 190°
21. cos 18° 129
22. sin 57° 409
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385
23. tan 88° 309
24. sin 105° 509
25. tan 172° 189
26. cos 205° 129
27. sin 205° 129
28. tan 266° 279
29. sec 72°
30. csc 15°
31. cot 63°
32. sec 100°
33. csc 125°
34. cot 165°
35. csc 245°
36. cot 254°
37. sec 307°
38. csc 347°
In 39–50, find the smallest positive value of u to the nearest degree.
39. sin u 5 0. 3455
40. cos u 5 0.4383
41. tan u 5 0.2126
42. cos u 5 0.7660
43. tan u 5 0.7000
44. sin u 5 0.9990
45. cos u 5 0.9990
46. tan u 5 1.8808
47. sin u 5 0.5446
48. cos u 5 0.5446
49. tan u 5 1.0355
50. tan u 5 12.0000
In 51–58, find the smallest positive value of u to the nearest minute.
51. sin u 5 0.2672
52. cos u 5 0.2672
53. sin u 5 0.9692
54. cos u 5 0.9692
55. sin u 5 0.6534
56. cos u 5 0.6534
57. tan u 5 7.3478
58. tan u 5 0.0892
Applying Skills
59. A ramp that is 12 feet long is used to reach a doorway that is 3.5 feet above the level
ground. Find, to the nearest degree, the measure the ramp makes with the ground.
60. The bed of a truck is 4.2 feet above the ground. In order to unload boxes from the truck, the
driver places a board that is 12 feet long from the bed of the truck to the ground. Find, to
the nearest minute, the measure the board makes with the ground.
61. Three roads intersect to enclose a small triangular park. A path that is 72 feet long extends
from the intersection of two of the roads to the third road. The path is perpendicular to that
road at a point 65 feet from one of the other intersections and 58 feet from the third. Find,
to the nearest ten minutes, the measures of the angles at which the roads intersect.
62. The terminal side of an angle in standard position intersects the unit circle at the point
(20.8, 0.6).
a. In what quadrant does the terminal side of the angle lie?
b. Find, to the nearest degree, the smallest positive measure of the angle.
63. The terminal side of an angle in standard position intersects the unit circle at the point
(0.28, 20.96).
a. In what quadrant does the terminal side of the angle lie?
b. Find, to the nearest degree, the smallest positive measure of the angle.
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9-8 REFERENCE ANGLES AND THE CALCULATOR
In Section 9-7, we used the calculator to find the function values of angles. We
know that two or more angles can have the same function value. For example,
when the degree measures of two angles differ by 360, these angles, in standard
position, have the same terminal side and therefore the same function values. In
addition to angles whose measures differ by a multiple of 360, are there other
angles that have the same function values?
Second-Quadrant Angles
In the diagram, R(1, 0) and P(a, b) are points
on the unit circle. Under a reflection in the
y-axis, the image of P(a, b) is P9(2a, b) and
the image of R(1, 0) is R9(21, 0). Since angle
measure is preserved under a line reflection,
mROP 5 mR9OP9.
h
y
P9(a, b)
P(a, b)
u
R9(1, 0)
O
180 2 u
x
R(1, 0)
h
The rays OR and ORr are opposite
rays and ROP9 and R9OP9 are supplementary. Therefore, ROP9 and ROP are
supplementary.
mROP9 1 mROP 5 180
mROP 5 180 2 mROP9
• If mROP9 5 u, then mROP 5 180 2 u.
• If mROP9 5 u, then sin u 5 b and cos u 5 2a.
• If mROP 5 (180 2 u), then sin (180 2 u) 5 b and cos (180 2 u) 5 a.
Therefore:
sin u 5 sin (180 2 u)
cos u 5 2cos (180 2 u)
sin u
Since tan u 5 cos
u , then also:
sin (180 2 u)
tan u 5 2 cos (180 2 u) 5 2tan (180 2 u)
Let u be the measure of a second-quadrant angle. Then there exists a firstquadrant angle with measure 180 2 u such that:
sin u 5 sin (180 2 u)
cos u 5 2cos (180 2 u)
tan u 5 2tan (180 2 u)
The positive acute angle (180 2 u) is the reference angle of the secondquadrant angle. When drawn in standard position, the reference angle is in the
first quadrant.
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387
EXAMPLE 1
Find the exact values of sin 120°, cos 120°, and tan 120°.
y
Solution The measure of the reference angle for an angle
of 120° is 180° 2 120° 5 60°.
sin 120° 5 sin 60° 5 !3
2
120°
60°
cos 120° 5 2cos 60° 5 212
O
x
tan 120° 5 2tan 60° 5 2!3
Third-Quadrant Angles
In the diagram, R(1, 0) and P(a, b) are two
points on the unit circle. Under a reflection in
the origin, the image of R(1, 0) is R9(21, 0) and
the image of P(a, b) is P9(2a, 2b). Since angle
measure is preserved under a point reflection,
mROP 5 mR9OP9.
R(1, 0)
y
u
O
P(a, b)
u 2 180
R(1, 0) x
P9(a, b)
mROP9 5 180 1 mR9OP9
5 180 1 mROP
mROP 5 mROP9 2 180
• If mROP9 5 u, then mROP 5 u 2 180.
• If mROP9 5 u, then sin u 5 2b and cos u 5 2a.
• If mROP 5 (u 2 180), then sin (u 2 180) 5 b and cos (u 2 180) 5 a.
Therefore:
sin u 5 2sin (u 2 180)
cos u 5 2 cos (u 2 180)
sin u
Since tan u 5 cos
u , then:
2 sin (u 2 180)
tan u 5 2 cos (u 2 180) 5 tan (u 2 180)
Let u be the measure of a third-quadrant angle. Then there exists a firstquadrant angle with measure u 2 180 such that:
sin u 5 2sin (u 2 180)
cos u 5 2cos (u 2 180)
tan u 5 tan (u 2 180)
The positive acute angle (u 2 180) is the reference angle of the third-quadrant
angle.When drawn in standard position, the reference angle is in the first quadrant.
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EXAMPLE 2
Express sin 200°, cos 200°, and tan 200° in terms of a function value of an acute
angle.
Solution An angle of 200° is in the third quadrant.
y
For a third-quadrant angle with measure u,
the measure of the reference angle is
u 2 180.
200°
O
20°
The measure of the reference angle for an
angle of 200° is 200° 2 180° 5 20°.
x
sin 200° 5 2sin 20°
cos 200° 5 2cos 20°
tan 200° 5 tan 20°
ENTER:
DISPLAY:
SIN
200
)
ENTER
COS
200
)
TAN
200
)
ENTER:
(-)
SIN
20
)
ENTER
ENTER
(-)
COS
20
)
ENTER
ENTER
TAN
sin(200)
-.3420201433
cos(200)
-.9396926208
tan(200)
.3639702343
DISPLAY:
20
)
ENTER
-sin(20)
-.3420201433
-cos(20)
-.9396926208
tan(20)
.3639702343
Answer sin 200° 5 2sin 20°, cos 200° 5 2cos 20°, tan 200° 5 tan 20°
EXAMPLE 3
If 180 , u , 270 and sin u 5 2 0.5726, what is the value of u to the nearest
degree?
Solution If mA 5 u, A is a third-quadrant angle. The measure of the reference angle
is u 2 180 and sin (u 2 180) 5 0.5726.
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ENTER:
DISPLAY:
2nd
SINⴚ1 0.5726
)
389
ENTER
s i n - 1( 0 . 5 7 2 6 )
34.9317314
To the nearest degree, (u 2 180) 5 35. Therefore:
u 5 35 1 180 5 215° Answer
Fourth-Quadrant Angles
In the diagram, R(1, 0) and P(a, b) are points
on the unit circle. Under a reflection in the
x-axis, the image of P(a, b) is P9(a, –b) and
the image of R(1, 0) is R(1, 0). Since angle
measure is preserved under a line reflection,
the measure of the acute angle ROP is
equal to the measure of the acute angle
ROP9.
y
P(a, b)
u
360 2 u
O
x
R(1, 0)
P(a, b)
• If mROP9 5 u, then mROP 5 360 2 u.
• If mROP9 5 u, then sin u 5 2b and cos u 5 a.
• If mROP 5 (360 2 u), then sin (360 2 u) 5 b and cos (360 2 u) 5 a.
Therefore:
sin u 5 2sin (360 2 u)
cos u 5 cos (360 2 u)
sin u
Since tan u 5 cos
u, then:
2 sin (360 2 u)
tan u 5 cos (360 2 u) 5 2tan (360 2 u)
Let u be the measure of a fourth-quadrant angle. Then there exists a firstquadrant angle with measure 360 2 u such that:
sin u 5 2sin (360 2 u )
cos u 5 cos (360 2 u)
tan u 5 2tan (360 2 u)
The positive acute angle (360 2 u) is the reference angle of the fourthquadrant angle. When drawn in standard position, the reference angle is in the
first quadrant.
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Trigonometric Functions
EXAMPLE 4
If sin u 5 20.7424, find, to the nearest degree, two positive values of u that are
less than 360°.
Solution Find the reference angle by finding arcsin 0.7424.
ENTER:
DISPLAY:
2nd
SINⴚ1 .7424
)
ENTER
s i n - 1( . 7 4 2 4 )
47.93626204
To the nearest degree, the measure of the reference angle is 48°. The sine is negative in the third quadrant and in the fourth quadrant.
In the third quadrant:
48 5 u 2 180
u 5 228°
In the fourth quadrant:
48 5 360 2 u
u 5 360 2 48
u 5 312°
Alternative u 5 the angle whose sine is 20.7424 and arcsin 20.7424 5 sin1 20.7424.
Solution
(-) .7424 )
SINⴚ1
2nd
ENTER
ENTER:
DISPLAY:
s i n - 1( - . 7 4 2 4 )
-47.93626204
There are many angles whose sine is 20.7424. The calculator returns the measure of the angle with the smallest absolute value. When the sine of the angle is
negative, this is an angle with a negative degree measure, that is, an angle
formed by a clockwise rotation. To find the positive measure of an angle with
the same terminal side, add 360: u 5 248 1 360 5 312°. This is a fourthquadrant angle whose reference angle is 360 2 312 5 48. There is also a thirdquadrant angle whose sine is 20.7424 and whose reference angle is 48. In the
third quadrant:
48 5 u 1 180
u 5 228°
Answer 228° and 312°
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391
SUMMARY
Let u be the measure of an angle 90° , u , 360°.
Second Quadrant
Third Quadrant
180 2 u
Reference Angle
u 2 180
Fourth Quadrant
360 2 u
sin u
sin (180 2 u)
2sin (u 2 180)
2sin (360 2 u)
cos u
–cos (180 2 u)
2cos (u 2 180)
cos (360 2 u)
tan u
–tan (180 2 u)
tan (u 2 180)
2tan (360 2 u)
Exercises
Writing About Mathematics
1. Liam said that if 0 , u , 90, then when the degree measure of a fourth-quadrant angle is
2u, the degree measure of the reference angle is u. Do you agree with Liam? Explain why
or why not.
2. Sammy said that if a negative value is entered for sin1, cos1, or tan1, the calculator will
return a negative value for the measure of the angle. Do you agree with Sammy? Explain
why or why not.
Developing Skills
In 3–7, for each angle with the given degree measure: a. Draw the angle in standard position.
b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the
x-axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle.
3. 120°
4. 250°
5. 320°
6. 245°
7. 405°
In 8–17, for each angle with the given degree measure, find the measure of the reference angle.
8. 100°
13. 310°
9. 175°
14. 95°
10. 210°
11. 250°
12. 285°
15. 290°
16. 2130°
17. 505°
In 18–27, express each given function value in terms of a function value of a positive acute angle
(the reference angle).
18. sin 215°
19. cos 95°
20. tan 255°
21. cos 312°
22. tan 170°
23. sin 285°
24. cos 245°
25. tan 305°
26. sin 256°
27. sin 500°
In 28–43, for each function value, if 0° u , 360°, find, to the nearest degree, two values of u.
28. sin u 5 0.3420
29. cos u 5 0.6283
30. tan u 5 0.3240
31. tan u 5 1.4281
32. sin u 5 0.8090
33. sin u 5 20.0523
34. cos u 5 20.3090
35. cos u 5 0.9205
36. tan u 5 29.5141
37. sin u 5 0.2419
38. cos u 5 20.7431
39. sin u 5 20.1392
40. tan u 5 20.1405
41. sin u 5 0
42. cos u 5 0
43. tan u 5 0
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CHAPTER SUMMARY
In ABC with a right angle at C, BC is the leg that is opposite A, AC is
the leg that is adjacent to A, and AB is the hypotenuse.
opp
sin A 5 hyp
adj
cos A 5 hyp
opp
tan A 5 adj
An angle is in standard position when its vertex is at the origin and its initial side is the nonnegative ray of the x-axis. The measure of an angle in standard
position is positive when the rotation from the initial side to the terminal side is
in the counterclockwise direction. The measure of an angle in standard position
is negative when the rotation from the initial side to the terminal side is in the
clockwise direction.
We classify angles in standard position according to the quadrant in which
the terminal side lies.
• If 0 , u , 90 and mAOB 5 u 1 360n for any integer n, then AOB is a
first-quadrant angle.
• If 90 , u , 180 and mAOB 5 u 1 360n for any integer n, then AOB is
a second-quadrant angle.
• If 180 , u , 270 and mAOB 5 u 1 360n for any integer n, then AOB
is a third-quadrant angle.
• If 270 , u , 360 and mAOB 5 u 1 360n for any integer n, then AOB
is a fourth-quadrant angle.
An angle in standard position whose terminal side lies on either the x-axis
or the y-axis is called a quadrantal angle. The measure of a quadrantal angle is
a multiple of 90.
Angles in standard position that have the same terminal side are coterminal
angles.
A circle with center at the origin and radius 1 is the unit circle and has the
equation x2 1 y2 5 1.
y
y
T(1, tan u)
P(cos u, sin u)
u
O
x
R(1, 0)
P(cos u, sin u)
u
O
x
R(1, 0)
T(1, tan u)
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Chapter Summary
y
393
y
T(1, tan u)
u
u
x
R(1, 0)
O
O
P(cos u, sin u)
x
R(1, 0)
P(cos u, sin u)
T(1, tan u)
Let P(p, q) be the point at which the terminal side of an angle in standard
position intersects the unit circle, and T(1, t) be the point at which the line
tangent to the circle at R(1, 0) intersects the terminal side of the angle. Let
mROP 5 u.
• The sine function is a set of ordered pairs (u, sin u) such that sin u 5 q.
• The cosine function is a set of ordered pairs (u, cos u) such that cos u 5 p.
• The tangent function is the set of ordered pairs (u, tan u) such that
tan u 5 t.
• The secant function is the set of ordered pairs (u, sec u) such that for all u
1
for which cos u 0, sec u 5 cos
u.
• The cosecant function is the set of ordered pairs (u, csc u) such that for all
1
u for which sin u 0, csc u 5 sin
u.
• The cotangent function is the set of ordered pairs (u, cot u) such that for
1
all u for which tan u is defined and not equal to 0, cot u 5 tan
u and for all u
for which tan u is undefined, cot u 5 0.
The secant, cosecant, and cotangent functions are called reciprocal functions.
The equilateral triangle and the isosceles right triangle make it possible to
find exact values for angles of 30°, 45°, and 60°.
u
0°
30°
sin u
0
cos u
1
tan u
0
1
2
!3
2
!3
3
45°
!2
2
!2
2
1
60°
!3
2
1
2
!3
90°
1
0
undefined
The rational approximation of the values of the sine, cosine, and tangent
functions can be displayed by most calculators.
If sin u 5 a, then u 5 arcsin a or u 5 sin1 a, read as “u is the angle whose
measure is a.” We can also write tan u 5 a as u 5 arctan a 5 tan1 a and
cos u 5 a as u 5 arccos a 5 cos 21 a.
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1
A minute is 60
of a degree. To express the measure of an angle in degrees
and minutes, multiply the decimal part of the degree measure by 60.
The trigonometric function values of angles with degree measure greater
than 90 or less than 0 can be found from their values at corresponding acute
angles called reference angles.
Second Quadrant Third Quadrant Fourth Quadrant
180 2 u
Reference Angle
u 2 180
360 2 u
sin u
sin (180 2 u)
2sin (u 2 180)
2sin (360 2 u)
cos u
2cos (180 2 u)
2cos (u 2 180)
cos (360 2 u)
tan u
2tan (180 2 u)
tan (u 2 180)
2tan (360 2 u)
VOCABULARY
9-1 Hypotenuse • Leg • Similar triangles • Sine • Cosine • Tangent
9-2 Initial side • Terminal side • Standard position • u (theta) • Quadrantal
angle • Coterminal angle • Angular speed
9-3 Unit circle • Sine function • Cosine function
9-4 Tangent function
9-5 Reciprocal function • Secant function • Cosecant function • Cotangent
function
9-7 Arcsine • Arccosine • Arctangent
9-8 Reference angle
REVIEW EXERCISES
In 1–5: a. Draw each angle in standard position. b. Draw its reference angle as
an acute angle formed by the terminal side of the angle and the x-axis. c. Draw
the reference angle in standard position. d. Give the measure of the reference
angle.
1. 220°
2. 300°
3. 145°
4. 2100°
5. 600°
6. For each given angle, find a coterminal angle with a measure of u such that
0° u , 360°: a. 505° b. 2302°
In 7–10, the terminal side of ROP in standard position intersects the unit circle
at P. If mROP is u, find:
a. the quadrant of ROP b. sin u c. cos u d. tan u e. csc u f. sec u g. cot u
7. P(0.8, 20.6)
8. P Q 34, !7
4 R
1
9. P Q22 !6
5 , 5R
10. PQ223, 2 !5
3 R
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395
In 11–14, the given point is on the terminal side of angle u in standard position.
Find: a. the quadrant of the angle b. sin u c. cos u d. tan u e. csc u f. sec u
g. cot u
11. (12, 9)
12. (2100, 0)
13. (28, 26)
14. (9, 213)
15. For each given angle in standard position, determine, to the nearest tenth,
the coordinates of the point where the terminal side intersects the unit circle: a. 405° b. 79°
In 16–27, find each exact function value.
16. cos 30°
17. sin 60°
18. tan 45°
19. tan 120°
20. cos 135°
21. sin 240°
22. cos 330°
23. sin 480°
24. sec 45°
25. csc 30°
26. cot 60°
27. sec 120°
In 28–35, find each function value to four decimal places.
28. cos 50°
29. tan 80°
30. sin 110°
31. tan 230°
32. cos 187°
33. sin (224°)
34. cos (2230°)
35. tan 730°
In 36–43, express each given function value in terms of a function value of a positive acute angle (the reference angle).
36. cos 100°
37. sin 300°
38. cos 280°
39. tan 210°
40. sin 150°
41. cos 255°
42. cos (240°)
43. tan (2310°)
In 44–51, for each function value, if 0 u , 360, find, to the nearest degree, two
values of u.
44. sin u 5 0.3747
45. cos u 5 0.9136
46. tan u 5 1.376
47. sin u 5 20.7000
48. tan u 5 20.5775
49. cos u 5 20.8192
50. sec u 5 1.390
51. csc u 5 3.072
In 52–55, 0 u , 360.
52. For what two values of u is tan u undefined?
53. For what two values of u is cot u undefined?
54. For what two values of u is sec u undefined?
55. For what two values of u is csc u undefined?
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56. In the diagram, mROP is u. Express each
of the following in terms of a coordinate of
R, P, or T.
a. sin u
b. cos u
y
T(1, t)
P(p, q)
c. tan u
u
O
d. Use similar triangles to show that
sin u
tan u 5 cos
u.
x
R(1, 0)
57. A road rises 630 feet every mile. Find to the nearest ten minutes the angle
that the road makes with the horizontal.
58. From the top of a building that is 56 feet high, the angle of depression to
the base of an adjacent building is 72°. Find, to the nearest foot, the distance between the buildings.
Exploration
h
In the diagram, OP intersects the unit circle at P.
y
Q(0, 1)
P
The line that is tangent to the circle at R(1, 0)
h
intersects OP at T. The line that is tangent to the
h
circle at Q(0,1) intersects OP at S. The measure
O
of ROP is u. Show that OT 5 sec u, OS 5 csc u,
u
S
T
x
R(1, 0)
and QS 5 cot u. (Hint: Use similar triangles.)
CUMULATIVE REVIEW
CHAPTERS 1–9
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. In simplest form, (2x 1 1)2 2 (x 1 2)2 is equal to
(1) 3x2 2 3
(3) 3x2 1 5
2
(2) 3x 1 8x 1 5
(4) 3x2 1 8x 2 3
2. The solution set of x2 2 5x 5 6 is
(1) {2, 3}
(2) {22, 23}
(3) {6, 21}
(4) {26, 1}
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Cumulative Review
3. In simplest form, the fraction
(1) 2a1
1
a2a
is equal to
a11
1
(2) a 2
a
1
(3) a 1
a
4. The sum of A3 1 !12B and A25 1 !27B is
(1) 16
397
(2) 22 1 !39
(3) 22 1 13 !3
(4) a 2 1
(4) 22 1 5 !3
5. Which of the following products is a rational number?
(1) A10 1 !10B A10 1 !10B
(2) A10 1 !10B A10 2 !10B
2 !3
6. The fraction 11 1
is equal to
!3
1
(1) 22
(2) 2
(3) !10A2 1 !10B
(4) 10A10 2 !10B
(3) 2 2 !3
(4) 22 1 !3
7. Which of the following is a one-to-one function when the domain is the set
of real numbers?
(1) y 5 x 2 5
(3) y 5 x2 2 2x 1 5
(2) x2 1 y2 5 9
(4) y 5 x 2 4
8. The sum of the roots of the equation 2x2 2 5x 1 3 5 0 is
(1) 5
(2) 25
(3) 25
9. When x and y vary inversely
(1) xy equals a constant.
(2) xy equals a constant.
(4) 252
(3) x 1 y equals a constant.
(4) x 2 y equals a constant.
10. The expression 2 log a 1 13 log b is equivalent to
(1) log 13a2b
(2) log 23ab
3
(3) log Aa2 1 !
bB
3
(4) log a2 A !
bB
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Express the product (3 2 2i)(21 1 i) in a 1 bi form.
12. Find the solution set of 2x 2 4 , 3.
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Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
1
13. Solve for x: 3 1 (x 1 3) 2 5 x.
14. Write the equation of the circle if the center of the circle is C(2, 1) and
one point on the circle is A(4, 0).
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. What are the roots of the equation x2 1 4 5 6x?
16. Let f(x) 5 x2 2 x and g(x) 5 5x 1 7.
a. Find f g(22).
b. Write h(x) 5 f g(x) as a polynomial in simplest form.
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CHAPTER
MORE
TRIGONOMETRIC
FUNCTIONS
The relationships among the lengths of the sides of
an isosceles right triangle or of the right triangles
formed by the altitude to a side of an equilateral triangle make it possible for us to find exact values of the
trig functions of angles of 30°, 45°, and their multiples.
How are the trigonometric function values for other
angles determined? Before calculators and computers
were readily available, handbooks that contained tables
of values were a common tool of mathematicians.
Now calculators or computers will return these values. How were the values in these tables or those displayed by a calculator determined?
In more advanced math courses, you will learn that
trigonometric function values can be approximated by
evaluating an infinite series to the required number of
decimal places.
3
5
10
CHAPTER
TABLE OF CONTENTS
10-1 Radian Measure
10-2 Trigonometric Function Values
and Radian Measure
10-3 Pythagorean Identities
10-4 Domain and Range of
Trigonometric Functions
10-5 Inverse Trigonometric
Functions
10-6 Cofunctions
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
7
x
x
x
sin x 5 x 2 3!
1 5!
2 7!
1c
To use this formula correctly, x must be expressed
in a unit of measure that has no dimension.This unit of
measure, which we will define in this chapter, is called
a radian.
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10-1 RADIAN MEASURE
A line segment can be measured in different units of measure such as feet,
inches, centimeters, and meters. Angles and arcs can also be measured in different units of measure. We have measured angles in degrees, but another frequently used unit of measure for angles is called a radian.
DEFINITION
A radian is the unit of measure of a central angle that intercepts an arc equal
in length to the radius of the circle.
2c
m
cm
1 cm
m
1c
1 1 1
In a circle of radius 1 centimeter, a central angle
1 cm
whose measure is 1 radian intercepts an arc whose
1
length is 1 centimeter.
1 cm
The radian measure of any central angle in a circle
of radius 1 is equal to the length of the arc that it intercepts. In a circle of radius 1 centimeter, a central angle of 2 radians intercepts
an arc whose length is 2 centimeters, and a central angle of 3 radians intercepts
an arc whose length is 3 centimeters.
6c
In a circle of radius 2 centimeters, a cen2 cm m
tral angle whose measure is 1 radian intercepts an arc whose length is 2 centimeters, a
central angle of 2 radians intercepts an arc
whose measure is 4 centimeters, and a cen1
1
1
tral angle of 3 radians intercepts an arc
2 cm
whose measure is 6 centimeters.
In general, the radian measure u of a
central angle of a circle is the length of the
intercepted arc, s, divided by the radius of
the circle, r.
m
2c
3 cm
1 cm
1
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Relationship Between Degrees and Radians
When the diameter of a circle is drawn, the straight
angle whose vertex is at the center of the circle intercepts an arc that is a semicircle. The length of the
semicircle is one-half the circumference of the circle
or 12 (2pr) 5 pr. Therefore, the radian measure of a
straight angle is pr
r 5 p. Since the degree measure of
a straight angle is 180°, p radians and 180 degrees
are measures of the same angle. We can write the following relationship:
p radians 5 180 degrees
pr
p
O
r
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Radian Measure
We will use this relationship to change the degree measure of any angle to
radian measure.
Changing Degrees to Radians
The measure of any angle can be expressed in degrees or in radians. We can do
this by forming the following proportion:
measure in degrees
measure in radians
180
5 p
This proportion can be used to find the radian measure of an angle whose
degree measure is known. For example, to find u, the radian measure of an angle
of 30°, substitute 30 for “measure in degrees” and u for “measure in radians” in
the proportion. Then use the fact that the product of the means is equal to the
product of the extremes to solve for u.
30
u
5 180
p
180u 5 30p
u 5 30p
180
u 5 p6
The radian measure of an angle of 30° is p6 .
We know that sin 30° 5 12. Use a calculator to compare this value to sin u
when u is p6 radians.
Begin by changing the calculator
to radian mode.
ENTER:
MODE
Then enter the expression.
ENTER:
MATH
ENTER
DISPLAY:
SIN
DISPLAY:
Normal
Sci Eng
Float 0123456789
Degree
Radian
The calculator confirms that sin 30° 5 sin p6 .
2nd
ENTER
ⴜ
p
6
ENTER
sin(π/6)Frac
1/2
)
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Changing Radians to Degrees
When the radian measure of an angle is known, we can use either of the two
methods, shown below, to find the degree measure of the angle.
METHOD 1 Proportion
Use the same proportion that was used to change from degrees to radians.
For example, to convert an angle of p2 radians to degrees, let u 5 degree measure of the angle.
(1) Write the proportion that relates degree
measure to radian measure:
measure in degrees
measure in radians
(2) Substitute u for the degree measure and
A p2 B
p
2
for the radian measure:
u
5 180
p
5 180
p
pu 5 p2 (180)
pu 5 90p
(3) Solve for u:
u 5 90
The measure of the angle is 90°.
METHOD 2 Substitution
Since we know that p radians 5 180°, substitute 180° for p radians and simplify the expression. For example:
p
2
radians 5 12(p radians) 5 12(180°) 5 90°
Radian measure is a ratio of the length of an arc to the length of the radius
of a circle.These ratios must be expressed in the same unit of measure.Therefore,
the units cancel and the ratio is a number that has no unit of measure.
EXAMPLE 1
The radius of a circle is 4 centimeters. A central angle of the circle intercepts an
arc of 12 centimeters.
a. What is the radian measure of the angle?
b. Is the angle acute, obtuse, or larger than a straight angle?
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Solution a.
How to Proceed
(1) Write the rule that states that the radian measure
of a central angle, u, is equal to the length s of the
intercepted arc divided by the length r of a radius:
u 5 sr
(2) Substitute the given values:
u 5 12
4
u53
(3) Simplify the fraction:
b. Since we know that p radians 5 180°, then 1 radian 5 1808
p .
3 radians 5 3 A 1808
p B 171.88°. Therefore, the angle is obtuse.
Answers a. 3 radians
b. Obtuse
EXAMPLE 2
The radius of a circle is 8 inches. What is the length of
the arc intercepted by a central angle of the circle if the
measure of the angle is 2.5 radians?
Solution
u5
2.5 5
2.5
8 in.
s
r
s
8
s 5 2.5(8) 5 20
Answer 20 inches
EXAMPLE 3
What is the radian measure of an angle of 45°?
How to Proceed
Solution
(1) Write the proportion:
(2) Substitute x for the radian
measure and 45 for the
degree measure:
(3) Solve for x:
measure in degrees
measure in radians
45
x
5 180
p
5 180
p
180x 5 45p
45
x 5 180
p
(4) Express x in simplest form:
Answer
1
4p
or p4
x 5 14p
45°
403
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EXAMPLE 4
The radian measure of an angle is 5p
6.
5p
6
a. What is the measure of the angle in degrees?
b. What is the radian measure of the reference angle?
Solution a. METHOD 1: Proportions
METHOD 2: Substitution
Let u 5 degree measure of
the angle.
Substitute 180° for p radians, and
simplify.
measure in degrees
measure in radians
5p
6
u
5p
6
180
5 p
5 180
p
pu 5
radians 5 56 (p radians)
5 56 (1808)
5 1508
180 A 5p
6 B
pu 5 150p
u 5 150
b. Since 5p
6 is a second-quadrant angle, its reference angle
is equal to 180° 2 150° or, in radians:
p
180° 2 150° → p 2 5p
6 5 6
Answers a. 150°
p
6
5p
6
b. p6
Exercises
Writing About Mathematics
1. Ryan and Rebecca each found the radian measure of a central angle by measuring the
radius of the circle and the length of the intercepted arc. Ryan used inches and Rebecca
used centimeters when making their measurements. If Ryan and Rebecca each measured
accurately, will the measures that they obtain for the angle be equal? Justify your answer.
2. If a wheel makes two complete revolutions, each spoke on the wheel turns through an angle
of how many radians? Explain your answer.
Developing Skills
In 3–12, find the radian measure of each angle whose degree measure is given.
3. 30°
4. 90°
5. 45°
6. 120°
7. 160°
8. 135°
9. 225°
10. 240°
11. 270°
12. 330°
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405
In 13–22, find the degree measure of each angle whose radian measure is given.
13. p3
14. p9
p
15. 10
16. 2p
5
17. 10p
9
18. 3p
2
19. 3p
20. 11p
6
21. 7p
2
22. 1
In 23–27, for each angle with the given radian measure: a. Give the measure of the angle in degrees.
b. Give the measure of the reference angle in radians. c. Draw the angle in standard position and its
reference angle as an acute angle formed by the terminal side of the angle and the x-axis.
23. p3
24. 7p
36
27. 25p
9
26. 27p
18
25. 10p
9
In 28–37, u is the radian measure of a central angle that intercepts an arc of length s in a circle with
a radius of length r.
28. If s 5 6 and r 5 1, find u.
29. If u 5 4.5 and s 5 9, find r.
30. If u 5 2.5 and r 5 10, find s.
31. If r 5 2 and u 5 1.6, find s.
32. If r 5 2.5 and s 5 15, find u.
33. If s 5 16 and u 5 0.4, find r.
34. If r 5 4.2 and s 5 21, find u.
35. If r 5 6 and u 5 2p
3 , find s.
36. If s 5 18 and u 5 6p
37. If u 5 6p and r 5 1, find s.
5 , find r.
38. Circle O has a radius of 1.7 inches. What is the length, in inches, of an arc intercepted by a
central angle whose measure is 2 radians?
39. In a circle whose radius measures 5 feet, a central angle intercepts an arc of length 12 feet.
Find the radian measure of the central angle.
40. The central angle of circle O has a measure of 4.2 radians and it intercepts an arc whose
length is 6.3 meters. What is the length, in meters, of the radius of the circle?
41. Complete the following table, expressing degree measures in radian measure in terms of p.
Degrees
30°
45°
60°
90°
180°
270°
360°
Radians
Applying Skills
42. The pendulum of a clock makes an angle of 2.5 radians as its tip travels 18 feet. What is the
length of the pendulum?
43. A wheel whose radius measures 16 inches is rotated. If a point on the circumference of the
wheel moves through an arc of 12 feet, what is the measure, in radians, of the angle through
which a spoke of the wheel travels?
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44. The wheels on a bicycle have a radius of 40 centimeters. The wheels on a cart have a radius
of 10 centimeters. The wheels of the bicycle and the wheels of the cart all make one complete revolution.
a. Do the wheels of the bicycle rotate through the same angle as the wheels of the cart?
Justify your answer.
b. Does the bicycle travel the same distance as the cart? Justify your answer.
45. Latitude represents the measure of a central angle with
vertex at the center of the earth, its initial side passing
through a point on the equator, and its terminal side passing through the given location. (See the figure.) Cities A
and B are on a north-south line. City A is located at 30°N
and City B is located at 52°N. If the radius of the earth is
approximately 6,400 kilometers, find d, the distance
between the two cities along the circumference of the
earth. Assume that the earth is a perfect sphere.
North
Pole
City B
City A
52°N
30°
6,400 km Equator
10-2 TRIGONOMETRIC FUNCTION VALUES AND RADIAN MEASURE
Using Radians to Find Trigonometric
Function Values
Hands-On Activity
In previous chapters, we found the exact trigonometric function values of angles
of 0o, 30o, 45o, 60o, 90o, 180°, and 270°. Copy and complete the table below to
show measures of these angles in radians and the corresponding function values. Let u be the measure of an angle in standard position.
u in Degrees
0°
30°
45°
60°
90°
180° 270°
u in Radians
sin u
cos u
tan u
We can use these function values to find the exact function value of any
angle whose radian measure is a multiple of p6 or p4 .
EXAMPLE 1
Find the exact value of tan 2p
3.
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Solution
METHOD 2
The radian measure 2p
3 is equivalent
y
p
3 (60°)
METHOD 1
2p
3 (120°)
x
to the degree measure
407
2(1808)
3
5 1208.
An angle of 120° is a second-quadrant
angle whose tangent is negative.
The reference angle is 180° 2 120° 5 60°.
2p
Since p2 , 2p
3 , p, an angle of 3
is a second-quadrant angle.
The reference angle of an angle
2p
p
of 2p
3 radians is p 2 3 5 3 .
The tangent of a second-quadrant
angle is negative.
tan 2p
3 5 tan 120° 5 2tan 60° 5 2!3
p
tan 2p
3 5 2tan 3 5 2!3
Answer 2!3
The graphing calculator will return the trigonometric function values of
angles entered in degree or radian measure. Before entering an angle measure,
be sure that the calculator is in the correct mode. For example, to find
tan p8 to four decimal places, use the following sequence of keys and round the
number given by the calculator. Be sure that your calculator is set to radian
mode.
ENTER:
TAN
)
p
2nd
ⴜ
8
DISPLAY:
ENTER
tan(π/8)
.4142135624
Therefore, tan p8 0.4142.
EXAMPLE 2
Find the value of cos u to four decimal places when u 5 2.75 radians.
Solution ENTER: COS 2.75
)
ENTER
DISPLAY:
Note that the cosine is negative because the
angle is a second-quadrant angle.
cos(2.75)
-.9243023786
Answer 20.9243
Finding Angle Measures in Radians
When the calculator is in radian mode, it will return the decimal value of the
radian measure whose function value is entered. The measure will not be in
terms of p.
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For instance, we know that tan 45° 5 1. Since 45 5 180
4 , an angle of 45° is an
angle of p4 radians. Use a calculator to find u in radians when tan u 5 1. Your calculator should be in radian mode.
ENTER:
TANⴚ1 1
2nd
)
ENTER
DISPLAY:
tan-1(1)
.7853981634
The value 0.7853981634 returned by the calculator is approximately equal
to p4 . We can verify this by dividing p by 4 and comparing the answers.
The exact radian measure of the angle is the irrational number p4 . The rational approximation of the radian measure of the angle is 0.7853981634.
EXAMPLE 3
Find, in radians, the smallest positive value of u if tan u 5 21.8762. Express the
answer to the nearest hundredth.
Solution The calculator should be in radian mode.
ENTER:
TANⴚ1 21.8762
2nd
)
DISPLAY:
ENTER
tan-1(-1.8762)
-1.081104612
y
Tangent is negative in the second and
fourth quadrants. The calculator has returned
the measure of the negative rotation in the
fourth quadrant. The angle in the second
quadrant with a positive measure is formed
by the opposite ray and can be found by
adding p, the measure of a straight angle, to
the given measure.
ENTER:
ANS
2nd
p
ⴙ
2nd
DISPLAY:
x
O
1
1.8762
Ans+π
2.060488041
ENTER
Check Verify this answer by finding tan 2.060488041.
ENTER:
TAN
ENTER
Answer 2.06
2nd
ANS
)
DISPLAY:
tan(Ans)
-1.8762
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409
EXAMPLE 4
Find, to the nearest ten-thousandth, a value of u in radians if csc u 5 2.369.
1
Solution If csc u 5 2.369, then sin u 5 2.369
.
ENTER:
SINⴚ1 1
2nd
)
ⴜ
DISPLAY:
2.369
ENTER
sin-1(1/2.369)
.4357815508
Rounded to four decimal places, u 5 0.4358. Answer
Exercises
Writing About Mathematics
1. Alexia said that when u is a second-quadrant angle whose measure is in radians, the measure
of the reference angle in radians is p 2 u. Do you agree with Alexia? Explain why or why not.
2. Diego said that when u is the radian measure of an angle, the angle whose radian measure is
2pn 1 u is an angle with the same terminal side for all integral values of n. Do you agree
with Diego? Explain why or why not.
Developing Skills
In 3–12, find the exact function value of each of the following if the measure of the angle is given in
radians.
3. sin p4
4. tan p3
5. cos p2
6. tan p6
8. sin 4p
3
9. tan 5p
4
10. sec p3
11. csc p
7. cos 2p
3
12. cot p4
In 13–24, find, to the nearest ten-thousandth, the radian measure u of a first-quadrant angle with the
given function value.
13. sin u 5 0.2736
14. cos u 5 0.5379
15. tan u 5 3.726
16. cos u 5 0.9389
17. sin u 5 0.8267
18. cos u 5 0.8267
19. tan u 5 1.5277
20. cot u 5 1.5277
21. sec u 5 5.232
22. cot u 5 0.3276
23. csc u 5 2.346
24. cot u 5 0.1983
25. If f(x) 5 sin A 13x B , find f A p2 B .
27. If f(x) 5 sin 2x 1 cos 3x, find f A p4 B .
26. If f(x) 5 cos 2x, find f A 3p
4 B.
28. If f(x) 5 tan 5x 2 sin 2x, find f A p6 B .
Applying Skills
29. The unit circle intersects the x-axis at R(1, 0) and the terminal side of ROP at P. What are
the coordinates of P if mRP 5 4.275?
X
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30. The x-axis intersects the unit circle at R(1, 0) and a circle
of radius 3 centered at the origin at A(3, 0). The terminal
side of ROP intersects the unit circle at P and the circle
of radius 3 at B. The measure of RP is 2.50.
y
B
X
P
a. What is the radian measure of ROP?
O
A(3, 0)
x
R(1, 0)
b. What is the radian measure of AOB?
c. What are the coordinates of P?
d. What are the coordinates of B?
e. In what quadrant do P and B lie? Justify your answer.
31. The wheels of a cart that have a radius of 12 centimeters move in a counterclockwise direction for 20 meters.
a. What is the radian measure of the angle through which each wheel has turned?
b. What is sine of the angle through which the wheels have turned?
32. A supporting cable runs from the ground to the top of a tree that is in danger of falling
down. The tree is 18 feet tall and the cable makes an angle of 2p
9 with the ground.
Determine the length of the cable to the nearest tenth of a foot.
p
33. An airplane climbs at an angle of 15
with the ground. When the airplane has reached an altitude of 500 feet:
a. What is the distance in the air that the airplane has traveled?
b. What is the horizontal distance that the airplane has traveled?
Hands-On Activity 1:The Unit Circle and Radian Measure
In Chapter 9, we explored trigonometric function values on the unit circle with regard to degree
measure. In the following Hands-On Activity, we will examine trigonometric function values on the
unit circle with regard to radian measure.
1. Recall that for any point P on the unit
h
circle, OP is the terminal side of an
angle in standard position. If the measure of this angle is u, the coordinates of
P are (cos u, sin u). Use your knowledge
of the exact cosine and sine values for
0°, 30°, 45°, 60°, and 90° angles to find
the coordinates of the first-quadrant
points.
y
p
p
2p (cos 2 , sin 2 )
,
sin
(cos 2p
)
3
3
(cos p3 , sin p3 )
p
3p
3p
p
(cos 4 , sin 4 )
cos p4 , sin p4 )
2
3 p(
4 (cos p , sin p)
, sin 5p
(cos 5p
6
6
6)
6
p
6
(cos p, sin p)
O
x
(cos 0, sin 0)
, sin 11p
(cos 11p
6 )
6
11p
7p
5p
5p
cos
,
sin
(
)
6
4
(cos 4 , sin 4 )
cos 5p
, sin 5p
(
)
4p, sin 4p
3
3
cos
(
3)
3
, sin 3p
(cos 3p
2)
2
, sin 7p
(cos 7p
6)
6
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Pythagorean Identities
411
2. The cosine and sine values of the other angles marked on the unit circle can be found by
relating them to the points in the first quadrant or along the positive axes. For example, the
2p
p
p
point A cos 2p
3 , sin 3 B is the image of the point A cos 3 , sin 3 B under a reflection in the
y-axis. Thus, if A cos p3 , sin p3 B 5 (a, b), then
A cos
2p
3,
sin 2p
3 B 5 (2a, b).
Determine the exact coordinates of the points shown on the unit circle to find the function
values.
Hands-On Activity 2: Evaluating the Sine and Cosine Functions
The functions sin x and cos x can be represented by the following series when x is in radians:
3
5
7
sin x 5 x 2 x3! 1 x5! 2 x7! 1 c
2
4
6
cos x 5 1 2 x2! 1 x4! 2 x6! 1 c
1. Write the next two terms for each series given above.
2. Use the first six terms of the series for sin x to approximate sin p4 to four decimal places.
3. Use the first six terms of the series for cos x to approximate cos p3 to four decimal places.
10-3 PYTHAGOREAN IDENTITIES
The unit circle is a circle of radius 1 with center
at the origin. Therefore, the equation of the unit
circle is x2 1 y2 5 12 or x2 1 y2 5 1. Let P be any
h
point on the unit circle and OP be the terminal
side of ROP, an angle in standard position whose
measure is u.The coordinates of P are (cos u, sin u).
Since P is a point on the circle whose equation is
x2 1 y2 5 1, and x 5 cos u and y 5 sin u:
y
P(cos u, sin u)
u
O
x
R(1, 0)
(cos u)2 1 (sin u)2 5 1
In order to emphasize that it is the cosine value and the sine value that are
being squared, we write (cos u)2 as cos2 u and (sin u)2 as sin2 u. Therefore, the
equation is written as:
cos2 u 1 sin2 u 5 1
This equation is called an identity.
DEFINITION
An identity is an equation that is true for all values of the variable for which
the terms of the variable are defined.
Because the identity cos2 u 1 sin2 u 5 1 is based on the Pythagorean theorem, we refer to it as a Pythagorean identity.
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EXAMPLE 1
Verify that cos2 p3 1 sin2 p3 5 1.
Solution We know that cos
p
3
5 12 and sin p3 5 !3
2 . Therefore:
2
cos u 1 sin2 u 5 1
2
2
Q 12 R 1 Q !3
2 R 5 1
1
4
1 34 5 1
1 51 ✔
We can write two related Pythagorean identities by dividing both sides of
the equation by the same expression.
Divide by sin2 u:
Divide by cos2 u:
sin u
Recall that tan u 5 cos
u and
u
Recall that cot u 5 tan1 u 5 cos
sin u and
sec u 5 cos1 u.
csc u 5 sin1 u.
For all values of u for which
cos u 0:
For all values of u for which
sin u 0:
cos2 u 1 sin2 u 5 1
cos2 u 1 sin2 u 5 1
cos2 u
sin2 u
sin2 u 1 sin2 u
u 2
A cos
sin u B 1 1
cos2 u
cos2 u
sin2 u
1
1 cos
2
u 5 cos2 u
2
sin u 2
1
1 1 A cos
u B 5 A cos u B
1 1 tan2 u 5 sec2 u
1 1 tan2 u 5 sec2 u
5 sin12 u
2
5 A sin1 u B
cot2 u 1 1 5 csc2 u
EXAMPLE 2
If cos u 5 13 and u is in the fourth quadrant, use an identity to find:
a. sin u
b. tan u
c. sec u
d. csc u
e. cot u
Solution Since u is in the fourth quadrant, cosine and its reciprocal, secant, are positive;
sine and tangent and their reciprocals, cosecant and cotangent, are negative.
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sin u
b. tan u 5 cos
u5
a. cos2 u 1 sin2 u 5 1
A 13 B
2
1
9
1 sin2 u 5 1
1 sin u 5 1
3
d. csc u 5 sin1 u 5
sin u 5 2#89
sin u 5 22 !2
3
b. 22 !2
1
3
5 22!2
c. sec u 5 cos1 u 5 11 5 3
2
sin2 u 5 89
Answers a. 22 !2
3
22 !2
3
413
c. 3
1
22 !2
3
3
5 22 !2
5 23 !2
4
e. cot u 5 tan1 u 5 221!2 5 2 !2
4
d. 23 !2
4
e. 2 !2
4
EXAMPLE 3
Write tan u 1 cot u in terms of sin u and cos u and write the sum as a single fraction in simplest form.
cos u
sin u
tan u 1 cot u 5 cos
u 1 sin u
Solution
cos u
sin u
cos u
sin u
5 cos
u 3 sin u 1 sin u 3 cos u
2u
cos2 u
5 cossinu sin
u 1 cos u sin u
2 u 1 cos2 u
5 sincos
u sin u
5 cos u1sin u Answer
SUMMARY
Reciprocal Identities
sec u 5
csc u 5
1
cos u
1
sin u
cot u 5 tan1 u
Quotient Identities
Pythagorean Identities
sin u
tan u 5 cos
u
cos u
cot u 5 sin u
cos2 u 1 sin2 u 5 1
1 1 tan2 u 5 sec2 u
cot2 u 1 1 5 csc2 u
Exercises
Writing About Mathematics
1. Emma said that the answer to Example 3 can be simplified to (sec u)(csc u). Do you agree
with Emma? Justify your answer.
2. Ethan said that the equations cos2 u 5 1 2 sin2 u and sin2 u 5 1 2 cos2 u are identities. Do
you agree with Ethan? Explain why or why not.
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Developing Skills
In 3–14, for each given function value, find the remaining five trigonometric function values.
3. sin u 5 15 and u is in the second quadrant.
4. cos u 5 43 and u is in the first quadrant.
5. cos u 5 234 and u is in the third quadrant.
6. sin u 5 223 and u is in the fourth quadrant.
7. sin u 5 23 and u is in the second quadrant.
8. tan u 5 22 and u is in the second quadrant.
9. tan u 5 4 and u is in the third quadrant.
10. sec u 5 28 and u is in the second quadrant.
11. cot u 5 53 and u is in the third quadrant.
12. csc u 5 45 and u is in the second quadrant.
13. sin u 5 45 and u is in the second quadrant.
14. cot u 5 26 and u is in the fourth quadrant.
In 15–22, write each given expression in terms of sine and cosine and express the result in simplest
form.
15. (csc u)(sin u)
16. (sin u)(cot u)
17. (tan u)(cos u)
18. (sec u)(cot u)
19. sec u 1 tan u
u
20. sec
csc u
cot u
21. csc2 u 2 tan
u
22. sec u (1 1 cot u) 2 csc u (1 1 tan u)
10-4 DOMAIN AND RANGE OF TRIGONOMETRIC FUNCTIONS
In Chapter 9, we defined the trigonometric
functions for degree measures on the unit
circle. Using arc length, we can define the
trigonometric functions on the set of real
numbers. Recall that the function values of
sine and cosine are given by the coordinates
of P, that is, P(cos u, sin u). The length of the
arc intercepted by the angle is given by
y
P(cos u, sin u)
s 5 ur
u
O
x
R(1, 0)
u 5 sr
or
s 5 ur
However, since the radius of the unit circle is 1, the length of the arc s is
equal to u. In other words, the length of the arc s is equivalent to the measure of
the angle u in radians, a real number. Thus, any trigonometric function of u is a
function on the set of real numbers.
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Domain and Range of Trigonometric Functions
The Sine and Cosine Functions
We have defined the sine function and the cosine function in terms of the measure of angle formed by a rotation. The measure is positive if the rotation is in
the counterclockwise direction and negative if the rotation is in the clockwise
rotation. The rotation can continue indefinitely in both directions.
The domain of the sine function and of the cosine function is the set of real
numbers.
To find the range, let P be any point on the unit
h
circle and OP be the terminal side of ROP, an
angle in standard position with measure u. The
coordinates of P are (cos u, sin u). Since the unit
circle has a radius of 1, as P moves on the circle, its
coordinates are elements of the interval [21, 1].
Therefore, the range of cos u and sin u is the set of
real numbers [21, 1].
To see this algebraically, consider the identity
cos2 u 1 sin2 u 5 1. Subtract cos2 u from both sides
of this identity and solve for sin u. Similarly, subtract sin2 u from both sides of this identity and
solve for cos u.
cos2 u 1 sin2 u 5 1
sin2 u 5 1 2 cos2 u
sin u 5 6"1 2 cos2 u
y
P
u
x
R(1, 0)
O
Q
cos2 u 1 sin2 u 5 1
cos2 u 5 1 2 sin2 u
cos u 5 6"1 2 sin2 u
The value of sin u will be a real number if and only if 1 2 cos2 u $ 0, that is,
if and only if cos u # 1 or 21 # cos u # 1.
The value of cos u will be a real number if and only if 1 2 sin2 u $ 0, that is,
if and only if sin u # 1 or 21 # sin u # 1.
The range of the sine function and of the cosine function is the set of real
numbers [21, 1], that is, the set of real numbers from 21 to 1, including 21
and 1.
The Secant Function
The secant function is defined for all real numbers such that cos u 0. We know
that cos u 5 0 when u 5 p2 , when u 5 3p
2 , and when u differs from either of these
values by a complete rotation, 2p.
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Therefore, cos u 5 0 when u equals:
p
2
p
2
p
2
p
2
1 2p
1 4p
1 6p
(
3p
2
3p
2
3p
2
3p
2
5 p2 1 p
1 2p 5 p2 1 3p
1 4p 5 p2 1 5p
1 6p 5 p2 1 7p
(
We can continue with this pattern, which can be summarized as p2 1 np for
all integral values of n. The domain of the secant function is the set of all real
numbers except those for which cos u 5 0.
The domain of the secant function is the set of real numbers except p
2 1 np
for all integral values of n.
The secant function is defined as sec u 5 cos1 u. Therefore:
When 0 , cos u # 1,
u
1
, cos
cos u # cos u
0
cos u
or
0 , 1 # sec u
When 21 # cos u , 0,
21
cos u
u
0
$ cos
cos u . cos u
or
2sec u $ 1 . 0
or
sec u # 21 , 0
The union of these two inequalities defines the range of the secant function.
The range of the secant function is the set of real numbers (2`, 21] : [1, `),
that is, the set of real numbers greater than or equal to 1 or less than or
equal to 21.
The Cosecant Function
The cosecant function is defined for all real numbers such that sin u 0. We
know that sin u 5 0 when u 5 0, when u 5 p, and when u differs from either of
these values by a complete rotation, 2p. Therefore, sin u 5 0 when u equals:
0
p
0 1 2p 5 2p
p 1 2p 5 3p
0 1 4p 5 4p
p 1 4p 5 5p
0 1 6p 5 6p
p 1 6p 5 7p
(
(
We can continue with this pattern, which can be summarized as np for all
integral values of n. The domain of the cosecant function is the set of all real
numbers except those for which sin u 5 0.
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417
The domain of the cosecant function is the set of real numbers except np
for all integral values of n.
The range of the cosecant function can be found using the same procedure
as was used to find the range of the secant function.
The range of the cosecant function is the set of real numbers
(2`, 21] : [1, `), that is, the set of real numbers greater than or equal to
1 or less than or equal to 21.
The Tangent Function
sin u
We can use the identity tan u 5 cos
u to determine the domain of the tangent
function. Since sin u and cos u are defined for all real numbers, tan u is defined for all real numbers for which cos u 0, that is, for all real numbers except
p
2 1 np.
The domain of the tangent function is the set of real numbers except
p
2
1 np for all integral values of n.
To find the range of the tangent function,
recall that the tangent function is defined on
the line tangent to the unit circle at R(1, 0).
In particular, let T be the point where the terminal side of an angle in standard position
intersects the line tangent to the unit circle at
R(1, 0). If mROT 5 u, then the coordinates
of T are (1, tan u). As the point T moves on the
tangent line, its y-coordinate can take on any
real-number value. Therefore, the range of
tan u is the set of real numbers.
To see this result algebraically, solve the
identity 1 1 tan2 u 5 sec2 u for tan2 u.
y
P T(1, tan u)
u
O
x
R(1, 0)
1 1 tan2 u 5 sec2 u
tan2 u 5 sec2 u 2 1
The range of the secant function is sec u $ 1 or sec u # 21. Therefore,
sec2 u $ 1. If we subtract 1 from each side of this inequality:
sec2 u 2 1 $ 0
tan2 u $ 0
tan u $ 0 or tan u # 0
The range of the tangent function is the set of all real numbers.
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The Cotangent Function
We can find the domain and range of the cotangent function by a procedure
similar to that used for the tangent function.
u
We can use the identity cot u 5 cos
sin u to determine the domain of the cotangent function. Since sin u and cos u are defined for all real numbers, cot u is
defined for all real numbers for which sin u 0, that is, for all real numbers
except np.
The domain of the cotangent function is the set of real numbers except np
for all integral values of n.
To find the range of the cotangent function, solve the identity
1 1 cot2 u 5 csc2 u for cot2 u.
1 1 cot2 u 5 csc2 u
cot2 u 5 csc2 u 2 1
The range of the cosecant function is csc u $ 1 or csc u # 21. Therefore,
csc2 u $ 1. If we subtract 1 from each side of this inequality:
csc2 u 2 1 $ 0
cot2 u $ 0
cot u $ 0 or cot u # 0
The range of the cotangent function is the set of all real numbers.
EXAMPLE 1
Explain why that is no value of u such that cos u 5 2.
Solution The range of the cosine function is 21 # cos u # 1. Therefore, there is no
value of u for which cos u is greater than 1.
SUMMARY
Function
Domain (n is an integer)
Range
Sine
All real numbers
[21, 1]
Cosine
All real numbers
[21, 1]
Tangent
All real numbers except p2 1 np
All real numbers except np
All real numbers
All real numbers except p2 1 np
All real numbers except np
(2`, 21] < [1, `)
Cotangent
Secant
Cosecant
All real numbers
(2`, 21] < [1, `)
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Exercises
Writing About Mathematics
1. Nathan said that cot u is undefined for the values of u for which csc u is undefined. Do you
agree with Nathan? Explain why or why not.
2. Jonathan said that cot u is undefined for p2 because tan u is undefined for p2 . Do you agree
with Nathan? Explain why or why not.
Developing Skills
In 3–18, for each function value, write the value or tell why it is undefined. Do not use a calculator.
3. sin p2
4. cos p2
5. tan p2
6. sec p2
7. csc p2
8. cot p2
9. tan p
10. cot p
11. sec 3p
2
12. csc 7p
2
13. tan 0
14. cot 0
15. tan A 2p2 B
16. csc A 29p
2 B
17. sec A 29p
2 B
18. cot (28p)
19. List five values of u for which sec u is undefined.
20. List five values of u for which cot u is undefined.
10-5 INVERSE TRIGONOMETRIC FUNCTIONS
Inverse Sine Function
We know that an angle of 30° is an angle of p6 radians and that an angle of 5(30°)
or 150° is an angle of 5 A p6 B or 5p
6 . Since 30 5 180 2 150, an angle of 30° is the reference angle for an angle of 150°. Therefore:
sin 30° 5 sin 150° 5 12
and
1
sin p6 5 sin 5p
6 52
1
Two ordered pairs of the sine function are A p6 , 12 B and A 5p
6 , 2 B . Therefore, the
sine function is not a one-to-one function. Functions that are not one-to-one do
not have an inverse function. When we interchange the elements of the ordered
pairs of the sine function, the set of ordered pairs is a relation that is not a function.
{(a, b) : b 5 sin a} is a function.
{(b, a) : a 5 arcsin b} is not a function.
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For every value of u such that 0 # u # p2 ,
there is exactly one nonnegative value of sin u.
For every value of u such that 2p2 # u , 0, that
is, for every fourth-quadrant angle, there is
exactly one negative value of sin u.
Therefore, if we restrict the domain of the
sine function to 2p2 # u # p2 , the function is
one-to-one and has an inverse function. We
designate the inverse sine function by arcsin or
sin21.
Sine Function with
a Restricted Domain
Domain 5 Ux :
p
2
O
x
2p2
Inverse Sine Function
y 5 arcsin x or y 5 sin21 x
y 5 sin x
2p2
y
#x#
pV
2
Range 5 {y : 21 # y # 1}
Domain 5 {x : 21 # x # 1}
Range 5 Uy : 2p2 # y # p2 V
Note that when we use the SINⴚ1 key on the graphing calculator with a
number in the interval [21, 1], the response will be an number in the interval
p p
S 22 , 2 T .
Inverse Cosine Function
The cosine function, like the sine function, is not a one-to-one function and does
not have an inverse function. We can also restrict the domain of the cosine function to form a one-to-one function that has an inverse function.
For every value of u such that 0 # u # p2 ,
y
there is exactly one nonnegative value of cos u.
For every value of u such that p2 , u # p, that is,
for every second-quadrant angle, there is exactly
p
one negative value of cos u. Therefore, if we
O
0
x
restrict the domain of the cosine function to
0 # u # p, the function is one-to-one and has an
inverse function. We designate the inverse
cosine function by arccos or cos21.
Cosine Function with
a Restricted Domain
Inverse Cosine Function
y 5 cos x
y 5 arccos x or y 5 cos21 x
Domain 5 {x : 0 # x # p}
Domain 5 {x : 21 # x # 1}
Range 5 {y : 21 # y # 1}
Range 5 {y : 0 # y # p}
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Note that when we use the COSⴚ1 key on the graphing calculator with a number in the interval [21, 1], the response will be an number in the interval [0, p].
Inverse Tangent Function
The tangent function, like the sine and cosine functions, is not a one-to-one function and does not have an inverse function. We can also restrict the domain of
the tangent function to form a one-to-one function that has an inverse function.
For every of u such that 0 # u , p2 , there is
y p
exactly one nonnegative value of tan u. For
2
every value of u such that 2p2 , u , 0, that is, for
every fourth-quadrant angle, there is exactly
O
one negative value of tan u. Therefore, if we
x
restrict the domain of the tangent function to
p
p
22 , u , 2 , the function is one-to-one and has
an inverse function. We designate the inverse
2p2
tangent function by arctan or tan21.
Tangent Function with
a Restricted Domain
Inverse Tangent Function
y 5 tan x
y 5 arctan x or y 5 tan21 x
Domain 5 U x : 2p2 , x , p2 V
Domain 5 {x : x is a real number}
Range 5 {y : y is a real number}
Range 5 U y : 2p2 , y , p2 V
Note that when we use the TANⴚ1 key on the graphing calculator with any
real number, the response will be an number in the interval A 2p2 , p2 B .
These same restrictions can be used to define inverse functions for the
secant, cosecant, and cotangent functions, which will be left to the student. (See
Exercises 40–42.)
EXAMPLE 1
Express the range of the arcsine function in degrees.
Solution Expressed in radians, the range of the arcsine function is S 2p2 , p2 T , that is,
2p2 # arcsin u # p2 . Since an angle of p2 radians is an angle of 90° and an angle
of 2p2 radians is an angle of 290°, the range of the arcsine function is:
[290°, 90°]
or
290° # u # 90° Answer
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EXAMPLE 2
Find sin (cos21 (20.5)).
Solution Let u 5 cos21 (20.5). Then u is a second-quadrant angle whose reference
angle, r, is r 5 p 2 u in radians or r 5 180 2 u in degrees.
In radians:
In degrees:
cos r 5 0.5
cos r 5 0.5
r 5 p3
p 2 u 5 p3
2p
3 5u
!3
sin u 5 sin 2p
3 5 2
r 5 60°
180° 2 u 5 60°
120° 5 u
sin u 5 sin 120° 5 !3
2
Alternative To find sin (cos21 (20.5)), find sin u when u 5 cos21 (20.5). If u 5 cos21
Solution (20.5), then cos u 5 20.5. For the function y 5 cos21 x, when x is negative,
p
2 , y # p. Therefore, u is the measure of a second-quadrant angle and sin u
is positive.
Use the Pythagorean identity:
sin2 u 1 cos2 u 5 1
sin2 u 1 (20.5)2 5 1
sin2 u 1 0.25 5 1
sin2 u 5 0.75
sin u 5 !0.75 5 !0.25!3 5 0.5 !3
Check ENTER: SIN
.5
)
2nd
2nd
)
¯ 3
COSⴚ1
(-)
ENTER
)
ⴜ
2
DISPLAY:
sin(cos-1(-.5))
.8660254038
√(3)/2
.8660254038
ENTER
Answer
!3
2
EXAMPLE 3
Find the exact value of sec A arcsin 23
5 B if the angle is a third-quadrant angle.
23
Solution Let u 5 A arcsin 23
5 B . This can be written as sin u 5 A 5 B . Then:
1
sec A arcsin 23
5 B 5 sec u 5 cos u
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423
We know the value of sin u. We can use a Pythagorean identity to find cos u:
cos u 5 6"1 2 sin 2 u
1
cos u
5
1
2
6"1 2 sin u
5
1
2
6#1 2 A 235 B
5
1
9
6#1 2 25
5
1
6#16
25
5 14 5 654
6
5
An angle in the third quadrant has a negative secant function value.
5
Answer sec A arcsin 23
5 B 5 24
Calculator ENTER: 1 ⴜ COS
Solution
23 ⴜ 5 )
ENTER
2nd
)
SINⴚ1
MATH
DISPLAY:
1/cos(sin-1(-3/5)
)Frac
5/4
ENTER
The calculator returns the value 54. The calculator used the function value of
A arcsin 23
5 B , a fourth-quadrant angle for which the secant function value is
positive. For the given third-quadrant angle, the secant function value is negative.
5
Answer sec A arcsin 23
5 B 5 24
Exercises
Writing About Mathematics
1. Nicholas said that the restricted domain of the cosine function is the same as the restricted
domain of the tangent function. Do you agree with Nicholas? Explain why or why not.
2. Sophia said that the calculator solution to Example 2 could have been found with the calculator set to either degree or radian mode. Do you agree with Sophia? Explain why or why
not.
Developing Skills
In 3–14, find each value of u: a. in degrees b. in radians
3. u 5 arcsin 12
6. u 5 arctan (21)
9. u 5 arctan A2!3B
12. u 5 arcsin 1
4. u 5 arctan 1
7. u 5 arccos A 212 B
10. u 5 arcsin Q2 !3
2 R
13. u 5 arctan 0
5. u 5 arccos 1
8. u 5 arcsin A 212 B
11. u 5 arccos (21)
14. u 5 arccos 0
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In 15–23, use a calculator to find each value of u to the nearest degree.
15. u 5 arcsin 0.6
16. u 5 arccos (20.6)
17. u 5 arctan 4.4
18. u 5 arctan (24.4)
19. u 5 arccos 0.9
20. u 5 arccos (20.9)
21. u 5 arcsin 0.72
22. u 5 arcsin (20.72)
23. u 5 arctan (217.3)
In 24–32, find the exact value of each expression.
24. sin (arctan 1)
25. cos (arctan 0)
28. tan A arcsin A 212 B
27. cos (arccos (21))
30. tan Qarccos Q2 !2
2 RR
26. tan (arccos 1)
B
29. cos Qarcsin Q2 !3
2 RR
31. sin Qarccos Q2 !2
2 RR
32. cos Qarcsin Q2 !2
2 RR
In 33–38, find the exact radian measure u of an angle with the smallest absolute value that satisfies
the equation.
33. sin u 5 !2
2
34. cos u 5 !3
2
35. tan u 5 1
36. sec u 5 21
37. csc u 5 2!2
38. cot u 5 !3
39. In a–c with the triangles labeled as shown, use the inverse trigonometric functions to
express u in terms of x.
b.
c.
1
1
3
a.
2x
x1
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u
u
x
x11
3
u
x12
Applying Skills
40. a. Restrict the domain of the secant function to form a one-to-one function that has an
inverse function. Justify your answer.
b. Is the restricted domain found in a the same as the restricted domain of the cosine
function?
c. Find the range of the restricted secant function.
d. Find the domain of the inverse secant function, that is, the arcsecant function.
e. Find the range of the arcsecant function.
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Cofunctions
41. a. Restrict the domain of the cosecant function to form a one-to-one function that has an
inverse function. Justify your domain.
b. Is the restricted domain found in a the same as the restricted domain of the sine function?
c. Find the range of the restricted cosecant function.
d. Find the domain of the inverse cosecant function, that is, the arccosecant function.
e. Find the range of the arccosecant function.
42. a. Restrict the domain of the cotangent function to form a one-to-one function that has an
inverse function. Justify your domain.
b. Is the restricted domain found in a the same as the restricted domain of the tangent
function?
c. Find the range of the restricted cotangent function.
d. Find the domain of the inverse cotangent function, that is, the arccotangent function.
e. Find the range of the arccotangent function.
43. Jennifer lives near the airport. An airplane approaching
the airport flies at a constant altitude of 1 mile toward a
point, P, above Jennifer’s house. Let u be the measure
of the angle of elevation of the plane and d be the horizontal distance from P to the airplane.
P
d
a. Express u in terms of d.
b. Find u when d 5 1 mile and when d 5 0.5 mile.
u
10-6 COFUNCTIONS
Two acute angles are complementary if the
sum of their measures is 90°. In the diagram,
ROP is an angle in standard position whose
measure is u and (p, q) are the coordinates of
h
P, the intersection of OP with the unit circle.
Under a reflection in the line y = x, the image
of (x, y) is (y, x). Therefore, under a reflection
in the line y = x, the image of P(p, q) is
P(q, p), the image of R(1, 0) is R(0, 1), and the
image of O(0, 0) is O(0, 0). Under a line reflection, angle measure is preserved. Therefore,
y
R9(0, 1) P9(q, p)
y5x
mROP 5 mROP 5 u
mROP 5 90° 2 mROP 5 90° 2 u
P(p, q)
u
u
O
R(1, 0)
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The two angles, ROP and ROP are in standard position. Therefore,
cos u 5 p
sin u 5 q
sin (90° 2 u) 5 p
cos (90° 2 u) 5 q
cos u 5 sin (90° 2 u)
sin u 5 cos (90° 2 u)
The sine of an acute angle is equal to the cosine of its complement. The sine
and cosine functions are called cofunctions. There are two other pairs of cofunctions in trigonometry.
q
p
u
cot u 5 cos
sin u 5 q
sin u
tan u 5 cos
u 5 p
cos (908 2 u)
q
sin (908 2 u)
p
cot (90° 2 u) 5 sin (908 2 u) 5 p
tan (90° 2 u) 5 cos (908 2 u) 5 q
tan u 5 cot (90° 2 u)
cot u 5 tan (90° 2 u)
The tangent and cotangent functions are cofunctions.
sec u 5 cos1 u 5 p1
csc u 5 sin1 u 5 q1
csc (90° 2 u) 5 sin (9081 2 u) 5 p1
1
1
sec (90° 2 u) 5 cos (908
2 u) 5 q
sec u 5 csc (90° 2 u)
csc u 5 sec (90° 2 u)
The secant and cosecant functions are cofunctions. Cofunctions represented in
radians are left to the student. (See Exercise 25.)
In any right triangle ABC, if C is the right angle, then A and B are
complementary angles. Therefore, mA 5 90 2 mB and mB 5 90 2 mA.
A
Sine and cosine are cofunctions: sin A 5 cos B and sin B 5 cos A.
C
B
Tangent and cotangent are cofunctions: tan A 5 cot B and tan B 5 cot A.
Secant and cosecant are cofunctions: sec A 5 csc B and sec B 5 csc A.
EXAMPLE 1
If tan 63.44° 5 2.00, find a value of u such that cot u 5 2.00.
Solution If u is the degree measure of a first-quadrant angle, then:
tan (90° 2 u) 5 cot u
tan 63.44° 5 cot u
Therefore, tan (90° 2 u) 5 tan 63.44 and 63.44 5 (90° 2 u).
63.44° 5 90° 2 u
u 5 90° 2 63.44°
5 26.56°
Answer u 5 26.56°
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EXAMPLE 2
Express sin 100° as the function value of an acute angle that is less than 45°.
Solution An angle whose measure is 100° is a second-quadrant angle whose reference
angle is 180° 2 100° or 80°.
Therefore, sin 100° 5 sin 80° 5 cos (90° 2 80°) 5 cos 10°.
Answer sin 100° 5 cos 10°
Exercises
Writing About Mathematics
1. Mia said that if you know the sine value of each acute angle, then you can find any trigonometric function value of an angle of any measure. Do you agree with Mia? Explain why or
why not.
2. If sin A 5 21, is cos A 5 !3
2 always true? Explain why or why not.
Developing Skills
In 3–22: a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the
value of the function value found in a.
3. sin 65°
4. cos 80°
5. tan 54°
6. sin 86°
7. csc 48°
8. sec 75°
9. cot 57°
10. cos 70°
11. sin 110°
12. tan 95°
13. cos 130°
14. sec 125°
15. sin 230°
16. cos 255°
17. tan 237°
18. csc 266°
19. cos 300°
20. sin 295°
21. cot 312°
22. sec 285°
23. If sin u 5 cos (20 1 u), what is the value of u?
24. For what value of x does tan (x 1 10) 5 cot (40 1 x)?
25. Complete the following table of cofunctions for radian values.
Cofunctions (degrees)
cos u 5 sin (90° 2 u)
sin u 5 cos (90° 2 u)
tan u 5 cot (90° 2 u)
cot u 5 tan (90° 2 u)
sec u 5 csc (90° 2 u)
csc u 5 sec (90° 2 u)
Cofunctions (radians)
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In 26–33: a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the
function value found in a.
26. sin p3
29. sec 2p
3
28. tan p6
31. cot p
27. cos p4
30. csc 5p
6
32. sin A 2p4 B
34. tan A 25p
3 B
33. cos 8p
3
CHAPTER SUMMARY
A radian is the unit of measure of a central angle that intercepts an arc
equal in length to the radius of the circle.
In general, the radian measure u of a central angle is the length of the intercepted arc, s, divided by the radius of the circle, r.
u 5 rs
The length of the semicircle is pr. Therefore, the radian measure of a
straight angle is pr
r 5 p. This leads to the relationship:
p radians 5 180 degrees
The measure of any angle can be expressed in degrees or in radians. We can
do this by forming the following proportion:
measure in degrees
measure in radians
5 180
p
When the calculator is in radian mode, it will return the decimal value of the
radian measure whose function value is entered. The measure will not be in
terms of p.
An identity is an equation that is true for all values of the variable for which
the terms of the variable are defined.
Reciprocal Identities
sec u 5
csc u 5
cot u 5
1
cos u
1
sin u
1
tan u
Quotient Identities
Pythagorean Identities
sin u
tan u 5 cos
u
cos u
cot u 5 sin u
cos2 u 1 sin2 u 5 1
1 1 tan2 u 5 sec2 u
cot2 u 1 1 5 csc2 u
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The domains and ranges of the six basic trigonometric functions are as
follows.
Function
Domain (n is an integer)
Range
Sine
All real numbers
[21, 1]
Cosine
All real numbers
[21, 1]
p
2
1 np
Tangent
All real numbers except
Cotangent
All real numbers except np
p
2
All real numbers
1 np
Secant
All real numbers except
Cosecant
All real numbers except np
All real numbers
(2`, 21] < [1, `)
(2`, 21] < [1, `)
The inverse trigonometric functions are defined for restricted domains.
Sine Function with
a Restricted Domain
Inverse Sine Function
y 5 sin x
y 5 arcsin x or y 5 sin21 x
Domain 5 Ux : 2p2 # x # p2 V
Range 5 {y : 21 # y # 1}
Domain 5 {x : 21 # x # 1}
Cosine Function with
a Restricted Domain
Inverse Cosine Function
y 5 cos x
y 5 arccos x or y 5 cos21 x
Domain 5 {x : 0 # x # p}
Domain 5 {x : 21 # x # 1}
Range 5 {y : 21 # y # 1}
Range 5 {y : 0 # y # p}
Tangent Function with
a Restricted Domain
Inverse Tangent Function
y 5 tan x
y 5 arctan x or y 5 tan21 x
Domain 5 U x : 2p2 , x , p2 V
Range 5 {y : y is a real number}
Range 5 Uy : 2p2 # y # p2 V
Domain 5 {x : x is a real number}
Range 5 U y : 2p2 , y , p2 V
In any right triangle ABC, if C is the right angle, then A and B are
complementary angles. Therefore, mA 5 90 2 mB and mB 5 90 2 mA.
A
• Sine and cosine are cofunctions: sin A 5 cos B and sin B 5 cos A.
• Tangent and cotangent are cofunctions: tan A 5 cot B and tan B 5 cot A.
C
B
• Secant and cosecant are cofunctions: sec A 5 csc B and sec B 5 csc A.
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VOCABULARY
10-1 Radian
10-3 Identity • Pythagorean identity
10-5 Restricted domain • Inverse sine function • Inverse cosine function •
Inverse tangent function
10-6 Cofunction
REVIEW EXERCISES
In 1–4, express each degree measure in radian measure in terms of p.
1. 75°
2. 135°
3. 225°
4. 260°
In 5–8, express each radian measure in degree measure.
5. p4
6. 2p
5
7. 7p
6
8. 2p8
9. What is the radian measure of a right angle?
In 10–17, in a circle of radius r, the measure of a central angle is u and the length
of the arc intercepted by the angle is s.
10. If r 5 5 cm and s 5 5 cm, find u.
11. If r 5 5 cm and u 5 2, find s.
12. If r 5 2 in. and s 5 3 in., find u.
13. If s 5 8 cm and u 5 2, find r.
14. If s 5 9 cm and r 5 3 mm, find u.
15. If s 5 p cm and u 5 p4 , find r.
16. If u 5 3 and s 5 7.5 cm, find r.
17. If u 5 p5 and r 5 10 ft, find s.
18. For what values of u (0° # u # 360°) is each of the following undefined?
a. tan u
b. cot u
c. sec u
d. csc u
19. What is the domain and range of each of the following functions?
a. arcsin u
b. arccos u
c. arctan u
In 20–27, find the exact value of each trigonometric function.
20. sin p6
21. cos p4
22. tan p3
24. cos 2p
3
25. tan 3p
4
26. sin 11p
6
23. sin p2
27. cos A 27p
6 B
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Cumulative Review
431
In 28–33, find the exact value of each expression.
3p
28. tan 3p
4 1 cot 4
5p
30. 2 tan 2p
3 2 3 sec 6
32. cot2 A p3 B 1 sec2 A p4 B
29. 2 csc p2 1 5 cot p2
2 7p
31. tan2 A 7p
6 B 2 sec A 6 B
33. 3 cot A p4 B ? 2 csc2 A p4 B
In 34–39, find the exact value of each expression.
34. sin Qarcsin2 !3
2 R
36. cot (arccos 21)
38. csc (arccot 21)
y
Q(0, 1)
P
u
O
x
R(1, 0)
35. tan Aarccot 2!3B
37. sec Qarcsin !2
2 R
39. cos (arcsec 2)
Exploration
In this exploration, we will continue the work begun in Chapter 9.
h
Let OP intersect the unit circle at P. The line that is tangent to the circle at
g
R(1, 0) intersects OP at T. The line that is tangent to the circle at Q(0, 1) interg
sects OP at S. The measure of ROP is u.
1. ROP is a second-quadrant angle as shown to the left. Locate points
S and T. Show that 2OT 5 sec u, OS 5 csc u, and 2QS 5 cot u.
2. Draw ROP as a third-quadrant angle. Let R(1, 0) and Q(0, 1) be fixed
points. Locate points S and T. Show that 2OT 5 sec u, 2OS 5 csc u, and
QS 5 cot u.
3. Draw ROP as a fourth-quadrant angle. Let R and Q be defined as
before. Locate points S and T. Show that OT sec u, OS csc u, and
QS cot u.
CUMULATIVE REVIEW
CHAPTERS 1–10
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. If sec u 5 !5, then cos u equals
(1) 2!5
(2) 2 !5
5
(3) !5
5
2. Which of the following is an irrational number?
(1) 0.34
(2) !121
(3) 3 1 2i
(4) 2 !5
5
(4) 2p
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More Trigonometric Functions
3. Which of the following is a function from {1, 2, 3, 4, 5} to {1, 2, 3, 4, 5} that
is one-to-one and onto?
(1) {(1, 2), (2, 3), (3, 4), (4, 5), (5, 5)}
(3) {(1, 3), (2, 3), (3, 3), (4, 3), (5, 3)}
(2) {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}
(4) {(1, 1), (2, 4), (3, 1), (4, 2), (5, 1)}
4. When written with a rational denominator, 3 11 !5 is equal to
(1) 3 114!5
5. The fraction
(2) 3 222!5
1 2 a1
1 2 a12
a2
(1) a 2
1
(3) 3 24 !5
(4) 3 18 !5
a2
(3) a 1
1
a
(4) a 1
1
is equal to
a
(2) a 2
1
6. The inverse function of the function y 5 2x 1 1 is
(1) y 5 21x 1 1
1
(3) y 5 x 2
2
1
(2) y 5 x 1
2
(4) y 5 22x 2 1
7. The exact value of cos 2p
3 is
(1) !3
2
(2) 2 !3
2
(4) 212
(3) 21
8. The series 22 1 2 1 6 1 10 1 ? ? ? written in sigma notation is
`
(1) a 22k
k51
`
(2) a (22 1 2k)
k51
`
(3) a (k 1 4k)
k51
`
(4) a (22 1 4(k 2 1))
k51
9. What are the zeros of the function y 5 x3 1 3x2 1 2x?
(1) 0 only
(3) 0, 1, and 2
(2) 22, 21, and 0
(4) The function has no zeros.
10. The product A !28B A !22B is equal to
(1) 4i
(2) 4
(3) 24
(4) 3i !2
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Express the roots of x2 2 6x 1 13 5 0 in a + bi form.
12. If 280 is the measure of an angle in degrees, what is the measure of the
angle in radians?
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Cumulative Review
433
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. What is the equation of a circle if the endpoints of the diameter are
(22, 4) and (6, 0)?
1
1
1 log13 243
14. Given log13 19 2 2 log13 27
:
a. Write the expression as a single logarithm.
b. Evaluate the logarithm.
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. If sec u 5 !3 and u is in the fourth quadrant, find cos u, sin u, and tan u in
simplest radical form.
16. Find graphically the common solutions of the equations:
y 5 x2 2 4x 1 1
y5x23
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CHAPTER
11
CHAPTER
TABLE OF CONTENTS
11-1 Graph of the Sine Function
11-2 Graph of the Cosine Function
11-3 Amplitude, Period, and Phase
Shift
11-4 Writing the Equation of a
Sine or Cosine Graph
11-5 Graph of the Tangent
Function
11-6 Graphs of the Reciprocal
Functions
11-7 Graphs of Inverse
Trigonometric Functions
11-8 Sketching Trigonometric
Graphs
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
434
GRAPHS OF
TRIGONOMETRIC
FUNCTIONS
Music is an integral part of the lives of most people. Although the kind of music they prefer will differ,
all music is the effect of sound waves on the ear. Sound
waves carry the energy of a vibrating string or column
of air to our ears. No matter what vibrating object is
causing the sound wave, the frequency of the wave
(that is, the number of waves per second) creates a
sensation that we call the pitch of the sound. A sound
wave with a high frequency produces a high pitch while
a sound wave with a lower frequency produces a lower
pitch. When the frequencies of two sounds are in the
ratio of 2 : 1, the sounds differ by an octave and produce a pleasing combination. In general, music is the
result of the mixture of sounds that are mathematically
related by whole-number ratios of their frequencies.
Sound is just one of many physical entities that are
transmitted by waves. Light, radio, television, X-rays,
and microwaves are others. The trigonometric functions that we will study in this chapter provide the
mathematical basis for the study of waves.
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435
11-1 GRAPH OF THE SINE FUNCTION
The sine function is a set of ordered pairs of real numbers. Each ordered pair
can be represented as a point of the coordinate plane. The domain of the sine
function is the set of real numbers, that is, every real number is a first element
of one pair of the function.
To sketch the graph of the sine function, we will plot a portion of the graph
using the subset of the real numbers in the interval 0 x 2p. We know that
sin p6 5 21 5 0.5
and that p6 is the measure of the reference angle for angles with measures of
5p 7p 11p
6 , 6 , 6 , . . . . We also know that
sin p3 5 !3
2 5 0.866025 . . .
and that p3 is the measure of the reference angle for angles with measures of
p
2p 4p 5p
3 , 3 , 3 , . . . . We can round the rational approximation of sin 3 to two decimal
places, 0.87.
x
0
p
6
p
3
p
2
2p
3
5p
6
p
7p
6
4p
3
3p
2
sin x
0
0.5
0.87
1
0.87
0.5
0
20.5
20.87
21
5p
3
11p
6
2p
20.87 20.5
0
y
1
O
p
6
p
3
p
2
2p
3
5p
6
p
7p
6
4p
3
3p
2
5p
3
11p
6
2p
x
–1
y = sin x
On the graph, we plot the points whose coordinates are given in the table.
Through these points, we draw a smooth curve. Note how x and y change.
• As x increases from 0 to p2 , y increases from 0 to 1.
• As x increases from p2 to p, y decreases from 1 to 0.
• As x increases from p to 3p
2 , y continues to decrease from 0 to 21.
• As x increases from 3p
2 to 2p, y increases from 21 to 0.
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Graphs of Trigonometric Functions
When we plot a larger subset of the domain of the sine function, this pattern is repeated. For example, add to the points given above the point whose
x-coordinates are in the interval 22p x 0.
x
sin x
22p 211p
6
0
0.5
25p
3
23p
2
24p
3
27p
6
2p
25p
6
22p
3
2p2
0.87
1
0.87
0.5
0
20.5
20.87
21
2p3
2p6
0
20.87 20.5
0
y
y 5 sin x
1
x
22p
23p
2
2p
2p2
O
21
p
2
p
3p
2
2p
Each time we increase or decrease the value of the x-coordinates by a multiple of 2p, the basic sine curve is repeated. Each portion of the graph in an
interval of 2p is one cycle of the sine function.
The graph of the function y 5 sin x is its own image under the translation
T2p,0. The function y 5 sin x is called a periodic function with a period of 2p
because for every x in the domain of the sine function, sin x 5 sin (x 1 2p).
y
1
2p
22p 23p
2
2p2
y = sin x
O
1
p
2
p
3p
2
2p
5p
2
3p
7p
2
x
The period of the sine function y 5 sin x is 2p.
Each cycle of the sine curve can be separated into four quarters. In the first
quarter, the sine curve increases from 0 to the maximum value of the function.
In the second quarter, it decreases from the maximum value to 0. In the third
quarter, it decreases from 0 to the minimum value, and in the fourth quarter, it
increases from the minimum value to 0.
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437
A graphing calculator will display the graph of the sine function.
STEP 1. Put the calculator in radian mode.
ENTER:
MODE
Normal Sci Eng
Float 0123456789
Radian Degree
ENTER
STEP 2. Enter the equation for the sine function.
ENTER:
Yⴝ
SIN
X,T,⍜,n
ENTER
STEP 3. To display one cycle of the curve, let
Plot1 Plot2 Plot3
\Y1 = sin(X
\Y2=
the window include values from 0 to 2p
for x and values slightly smaller than
21 and larger than 1 for y. Use the following viewing window: Xmin 5 0,
Xmax 5 2p, Xscl 5 p6 , Ymin 5 21.5,
Ymax 5 1.5. (Note: Xscl changes the
scale of the x-axis.)
WINDOW
Xmin=0
Xmax=6.2831853...
Xscl=.52359877...
Ymin=-1.5
Ymax=1.5
Yscl=1
Xres=1
ENTER: WINDOW 0 ENTER
ENTER
p
ⴜ
2 2nd
p
2nd
6 ENTER 21.5 ENTER 1.5 ENTER
STEP 4. Finally, graph the sin curve by pressing GRAPH . To display more than
one cycle of the curve, change Xmin or Xmax of the window.
ENTER: WINDOW 22
ENTER
2nd
GRAPH
p
ENTER
4 2nd
p
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Graphs of Trigonometric Functions
EXAMPLE 1
In the interval 22p x 0, for what values of x does y 5 sin x increase and for
what values of x does y 5 sin x decrease?
Solution The graph shows that y 5 sin x
increases in the interval
22p x 23p
2 and in the interval
2p2 x 0 and decreases in the
p
interval 23p
2 x 22 .
y
1
x
22p
23p
2
2p
2p2
O
21
The Graph of the Sine Function and the Unit Circle
y
1
P(p, q)
A(u, q)
1
u
O
21
O
R
p
2p
x
21
21
Recall from Chapter 9 that if ROP is an angle in standard position with measure u and P(p, q) is a point on the unit circle, then (p, q) 5 (cos u, sin u) and
A(u, q) is a point on the graph of y 5 sin x. Note that the x-coordinate of A on
the graph of y 5 sin x is u, the length of RP.
Compare the graph of the unit circle and the graph of y 5 sin x in the figures below for different values of u.
X
5p
6
2p
3
p
2
p
3
y
p
6
7p 4p 3p 5p 11p
6 3 2
3
6
0
p
7p
6
y
O
4p
3
x
11p
5p 6
3p 3
2
O
p
6
p
3
p
2
2p 5p
3 6
p
x
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439
Hands-On Activity: Unwrapping the Unit Circle
We can use the graphing calculator to explore the unit circle and its relationship
to the sine and cosine functions.
STEP 1. Press MODE . Select RADIAN
Normal Sci Eng
mode, PAR graphing mode, and
Float 0123456789
SIMUL graphing mode.
Radian Degree
STEP 2. Press WINDOW to enter the window
Func Par Pol Seq
Connected Dot
screen. Use the following viewing
S
equential Simul
window:
Real a+bi re^ui
Full Horiz G-T
Tmin 5 0, Tmax 5 2p, Tstep 5 0.1, Xmin 5 21, Xmax 5 2p,
Xscl 5 p6 , Ymin 5 22.5, Ymax 5 2.5
WINDOW
Ts t e p = . 1
Xmin=-1
Xmax=6.2831853...
Xscl=.52359877...
Ymin=-2.5
Ymax=2.5
Yscl=1
<
WINDOW
Tmin=0
Tmax=6.2831853...
Ts t e p = . 1
Xmin=-1
Xmax=6.2831853...
Xscl=.52359877...
Ymin=-2.5
<
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Use the
and
arrow keys to display Ymin and Ymax.
STEP 3. Recall from Chapter 9 that a point P
Plot1 Plot2 Plot3
\X1T = cos(T)
Y1T = sin(T)
\X2T = T
Y2T =sin(T)
\X3T =
Y3T =
\X4T =
on the unit circle has coordinates
(cos u, sin u) where u is the measure of
the standard angle with terminal side
through P. We can define a function on
the graphing calculator that consists of
the set of ordered pairs (cos u, sin u).
Its graph will be the unit circle.
ENTER:
Yⴝ
COS
X,T,⍜,n
)
SIN
X,T,⍜,n
)
This key sequence defines the function consisting of the set of ordered
pairs (cos T, sin T). The variable T represents u on the graphing calculator.
STEP 4. Similarly, we can define a function consisting of the set of ordered pairs
(u, sin u).
ENTER:
X,T,⍜,n
SIN
X,T,⍜,n
)
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Graphs of Trigonometric Functions
STEP 5. Press GRAPH to watch the unit circle
“unwrap” into the sine function. As
the two functions are plotted, press
u
ENTER to pause and resume the
u
animation. You will see the unit circle
P
R
swept out by point P (represented by
a dot). Simultaneously, the graph of
y 5 sin u will be plotted to the right of
the y-axis by a point R. Notice that as
point P is rotated about the unit circle, the x-coordinate of R is equal
to u, the length of the arc from the positive ray of the x-axis to P. The
y-coordinate of R is equal to the y-coordinate of P.
Exercises
Writing About Mathematics
1. Is the graph of y 5 sin x symmetric with respect to a reflection in the origin? Justify your
answer.
2. Is the graph of y 5 sin x symmetric with respect to the translation T–2p,0? Justify your answer.
Developing Skills
3. Sketch the graph of y 5 sin x in the interval 0 x 4p.
a. In the interval 0 x 4p, for what values of x is the graph of y 5 sin x increasing?
b. In the interval 0 x 4p, for what values of x is the graph of y 5 sin x decreasing?
c. How many cycles of the graph of y 5 sin x are in the interval 0 x 4p?
4. What is the maximum value of y on the graph of y 5 sin x?
5. What is the minimum value of y on the graph of y 5 sin x?
6. What is the period of the sine function?
7. Is the sine function one-to-one? Justify your answer.
8. a. Point P is a point on the unit circle. The y-coordinate of P is sin p3 . What is the x-coordinate of P?
b. Point A is a point on the graph y 5 sin x. The y-coordinate of A is sin p3 . What is the
x-coordinate of A?
Applying Skills
9. A function f is odd if and only if f(x) 5 2f(2x) for all x in the domain of the function. Note
that a function is odd if it is symmetric with respect to the origin. In other words, the function is its own image under a reflection about the origin.
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441
a. Draw a unit circle and any first-quadrant angle ROP in standard position, with point P
on the unit circle. Let mROP 5 u.
b. On the same set of axes, draw an angle in standard position with measure 2u. What is
the relationship between u and 2u? Between sin u and sin (2u)?
c. Repeat steps a and b for second-, third-, and fourth-quadrant angles. Does
sin u 5 2sin (2u) for second-, third-, and fourth-quadrant angles? Justify your answer.
d. Does sin u 5 2sin (2u) for quadrantal angles? Explain.
e. Do parts a–d show that y 5 sin x is an odd function? Justify your answer.
10. City firefighters are told that they can use their
25-foot long ladder provided the measure of the
angle that the ladder makes with the ground is at
least 15° and no more than 75°.
a. If u represents the measure of the angle that
the ladder makes with the ground in radians,
what is a reasonable set of values for u?
Explain.
h
u
b. Express as a function of u, the height h of the
point at which the ladder will rest against a
building.
c. Graph the function from part b using the set of values for u from part a as the domain
of the function.
d. What is the highest point that the ladder is allowed to reach?
11. In later courses, you will learn that the sine function can be written as the sum of an infinite
sequence. In particular, for x in radians, the sine function can be approximated as the finite
series:
3
5
sin x x 2 x3! 1 x5!
3
5
a. Graph Y1 5 sin x and Y2 5 x 2 x3! 1 x5! on the graphing calculator. For what values of
x does Y2 seem to be a good approximation for Y1?
7
b. The next term of the sine approximation is 2x7! . Repeat part a using Y1 and
3
5
7
Y3 5 x 2 x3! 1 x5! 2 x7! . For what values of x does Y3 seem to be a good approximation
for Y1?
c. Use Y2 and Y3 to find approximations to the sine function values below. Which function gives a better approximation? Is this what you expected? Explain.
(1) sin p6
(2) sin p4
(3) sin p
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11-2 GRAPH OF THE COSINE FUNCTION
The cosine function, like the sine function, is a set of ordered pairs of real numbers. Each ordered pair can be represented as a point of the coordinate plane.
The domain of the cosine function is the set of real numbers, that is, every real
number is a first element of one pair of the function.
To sketch the graph of the cosine function, we plot a portion of the graph
using a subset of the real numbers in the interval 0 x 2p. We know that
cos p6 5 !3
2 5 0.866025 . . .
and that p6 is the measure of the reference angle for angles with measures of 5p
6,
7p 11p
6, 6 ,
. . . . We can round the rational approximation of cos p6 to two decimal
places, 0.87.
We also know that
cos p3 5 12 5 0.5
and that p3 is the measure of the reference angle for angle with measures of 2p
3,
4p 5p
3 , 3 ,.
...
x
0
p
6
p
3
p
2
2p
3
5p
6
p
7p
6
4p
3
3p
2
5p
3
11p
6
2p
cos x
1
0.87
0.5
0
20.5
20.87
21
20.87
20.5
0
0.5
0.87
1
5p
6
p
7p
6
y
1
O
p
6
p
3
p
2
2p
3
4p
3
3p
2
5p
3
11p
6
2p
x
–1
y = cos x
On the graph, we plot the points whose coordinates are given in the table.
Through these points, we draw a smooth curve. Note how x and y change.
• As x increases from 0 to p2 , y decreases from 1 to 0.
• As x increases from p2 to p, y decreases from 0 to 21.
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443
• As x increases from p to 3p
2 , y increases from 21 to 0.
• As x increases from 3p
2 to 2p, y increases from 0 to 1.
When we plot a larger subset of the domain of the cosine function, this pattern is repeated. For example, add to the points given above the point whose
x-coordinates are in the interval 22p x 0.
x
cos x
22p 211p
6
1
0.87
25p
3
23p
2
24p
3
27p
6
2p
25p
6
22p
3
2p2
2p3
2p6
0
0.5
0
20.5
20.87
21
20.87
20.5
0
0.5
0.87
1
y
y 5 cos x
1
x
22p
23p
2
2p
2p2
O
p
p
2
21
3p
2
2p
Each time we change the value of the x-coordinates by a multiple of 2p, the
basic cosine curve is repeated. Each portion of the graph in an interval of 2p is
one cycle of the cosine function.
The graph of the function y 5 cos x is its own image under the translation
T2p,0. The function y 5 cos x is a periodic function with a period of 2p because
for every x in the domain of the cosine function, cos x 5 cos (x 1 2p).
y
y 5 cos x
22p 23p
2
2p
2p2
O
–1
p
2
p
3p
2
2p
5p
2
x
3p
The period of the cosine function y 5 cos x is 2p.
Each cycle of the cosine curve can be separated into four quarters. In the
first quarter, the cosine curve decreases from the maximum value of the function to 0. In the second quarter, it decreases from 0 to the minimum value. In the
third quarter, it increases from the minimum value to 0, and in the fourth quarter, it increases from 0 to the maximum value.
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A graphing calculator will display the graph of the cosine function.
STEP 1. Put the calculator in radian mode.
STEP 2. Enter the equation for the cosine
Plot1 Plot2 Plot3
\Y1 = cos(X
\Y2=
function.
ENTER:
Yⴝ
COS
X,T,⍜,n
ENTER
STEP 3. To display one cycle of the curve,
WINDOW
Xmin=0
Xmax=6.2831853...
Xscl=.52359877...
Ymin=-1.5
Ymax=1.5
Yscl=1
Xres=1
let the window include values from
0 to 2p for x and values slightly
smaller than 21 and larger than 1 for
y. Use the following viewing window:
Xmin 5 0, Xmax 5 2p, Xscl 5 p6 ,
Ymin 5 21.5, Ymax 5 1.5.
ENTER:
WINDOW 0 ENTER
2nd
p
ⴜ
2 2nd
p
ENTER
6 ENTER 21.5 ENTER 1.5 ENTER
STEP 4. Finally, graph the sin curve by pressing GRAPH . To display more than
one cycle of the curve, change Xmin or Xmax of the window.
ENTER:
WINDOW 22
ENTER
2nd
p
ENTER
4 2nd
p
GRAPH
EXAMPLE 1
For what values of x in the interval 0 x 2p does y 5 cos x have a maximum
value and for what values of x does it have a minimum value?
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Graph of the Cosine Function
Solution The graph shows that y 5 cos x has
a maximum value, 1, at x 5 0 and
at x 5 2p and has a minimum
value, 21, at x 5 p.
1
y
x
O
p
2
p
3p
2
2p
21
The Graph of the Cosine Function
and the Unit Circle
y
1
P(p, q)
1
A(u, q)
u
R
O
21
O
2p x
p
21
21
If ROP is an angle in standard position with measure u and P(p, q) is a
point on the unit circle, then (p, q) 5 (cos u, sin u) and A(u, p) is a point on the
graph of y 5 cos x. Note that the x-coordinate of A on the graph of y 5 cos x
is u, the length of RP.
Compare the graph of the unit circle and the graph of y 5 cos x in the figures below for different values of u.
X
5p
6
2p
3
y
p
3
p
6
2p 5p
3
6
0
p
O
7p
6
p
2
4p
3
x
11p
5p 6
3p 3
2
O
p
6
p
3
p
2
p
7p 4p
6 3
3p 5p 11p
2 3 6
Exercises
Writing About Mathematics
1. Is the graph of y 5 cos x its own image under a reflection in the y-axis? Justify your answer.
2. Is the graph of y 5 cos x its own image under the translation T–2p,0? Justify your answer.
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Developing Skills
3. Sketch the graph of y 5 cos x in the interval 0 x 4p.
a. In the interval 0 x 4p, for what values of x is the graph of y 5 cos x increasing?
b. In the interval 0 x 4p, for what values of x is the graph of y 5 cos x decreasing?
c. How many cycles of the graph of y 5 cos x are in the interval 0 x 4p?
4. What is the maximum value of y on the graph of y 5 cos x?
5. What is the minimum value of y on the graph of y 5 cos x?
6. What is the period of the cosine function?
7. Is the cosine function one-to-one? Justify your answer.
Applying Skills
8. A function f is even if and only if f(x) 5 f(2x) for all x in the domain of the function. Note
that a function is even if it is symmetric with respect to the y-axis. In other words, a function
is even if it is its own image under a reflection about the y-axis.
a. Draw a unit circle and any first-quadrant angle ROP in standard position, with point P
on the unit circle. Let mROP 5 u.
b. On the same set of axes, draw an angle in standard position with measure 2u. What is the
relationship between u and 2u? Between cos u and cos (2u)?
c. Repeat steps a and b for second-, third-, and fourth-quadrant angles. Does cos u 5 cos (2u)
for second-, third-, and fourth-quadrant angles? Justify your answer.
d. Does cos u 5 cos (2u) for quadrantal angles? Explain.
e. Do parts a–d show that y 5 cos x is an even function? Justify your answer.
9. A wheelchair user brings along a 6-foot long portable ramp to get into a van. For safety and
ease of wheeling, the ramp should make a 5- to 10-degree angle with the ground.
a. Let u represent the measure of the angle that the ramp makes with the ground in radians.
Express, as a function of u, the distance d between the foot of ramp and the base of the
van on which the ramp sits.
b. What is the domain of the function from part a?
c. Graph the function from part a using the domain found in part b.
d. What is the smallest safe distance from the foot of the ramp to the base of the van?
10. In later courses, you will learn that the cosine function can be written as the sum of an infinite sequence. In particular, for x in radians, the cosine function can be approximated by the
finite series:
2
4
cos x 1 2 x2! 1 x4!
2
4
a. Graph Y1 5 cos x and Y2 5 1 2 x2! 1 x4! on the graphing calculator. For what values of x
does Y2 seem to be a good approximation for Y1?
6
b. The next term of the cosine approximation is 2x6! . Repeat part a using Y1 and
2
4
6
Y3 5 1 2 x2! 1 x4! 2 x6! . For what values of x does Y3 seem to be a good approximation
for Y1?
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447
c. Use Y2 and Y3 to find approximations to the cosine function values below. Which
function gives a better approximation? Is this what you expected? Explain.
(1) cos 2p6
(2) cos 2p4
(3) cos 2p
11-3 AMPLITUDE, PERIOD, AND PHASE SHIFT
In Chapter 4 we saw that the functions af(x), f(ax), and f(x) 1 a are transformations of the function f(x). Each of these transformations can be applied to the
sine function and the cosine function.
Amplitude
How do the functions y 5 2 sin x and y 5 12 sin x compare with the function
y 5 sin x? We will make a table of values and sketch the curves. In the following table, approximate values of irrational values of the sine function are used.
x
0
p
6
p
3
p
2
2p
3
5p
6
p
7p
6
4p
3
3p
2
sin x
0
0.5
0.87
1
0.87
0.5
0
20.5
20.87
21
20.87 20.5
0
2 sin x
0
1
1.73
2
1.73
1
0
21
21.73
22
21.73
0
1
2
0
0.25
0.43
0.5
0.43
0.25
0
sin x
20.25 20.43 20.5
11p
6
5p
3
21
2p
20.43 20.25
0
11p
6
2p x
y
2
y = 2 sin x
y = sin x
1
y=
O
–1
–2
p
6
p
3
1
2
p
2
sin x
2p
3
5p
6
p
7p
6
4p
3
3p
2
5p
3
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When we use these values to sketch the curves y 5 sin x, y 5 2 sin x, and
y 5 12 sin x, we see that y 5 2 sin x is the function y 5 sin x stretched in the vertical direction and y 5 21 sin x is the function y 5 sin x compressed in the vertical direction as expected.
• For y 5 sin x, the maximum function value is 1 and the minimum function
value is 21.
• For y 5 2 sin x, the maximum function value is 2 and the minimum function value is 22.
• For y 5 12 sin x, the maximum function value is 12 and the minimum function value is 212.
In general:
For the function y 5 a sin x, the maximum function value is a and the
minimum function value is 2a.
This is also true for the function y 5 a cos x.
For the function y 5 a cos x, the maximum function value is a and the
minimum function value is 2a.
The amplitude of a periodic function is the absolute value of one-half the
difference between the maximum and minimum y-values.
• For y 5 sin x, the amplitude is P
• For y 5
• For y 5
1 2 (21)
P 5 1.
2
2 2 (22)
P 5 2.
2 sin x, the amplitude is P
2
1 2 21
A 2B † 1
1
† 2
5 2.
2 sin x, the amplitude is
2
In general:
For y 5 a sin x and y 5 a cos x, the amplitude is P
a 2 (2a)
2
P 5 a.
EXAMPLE 1
For the function y 5 3 cos x:
a. What are the maximum and minimum values of the function?
b. What is the range of the function?
c. What is the amplitude of the function?
Solution The range of the function y 5 cos x is 21 y 1.
The function y 5 3 cos x is the function y 5 cos x stretched by a factor of 3 in
the vertical direction.
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449
When x 5 0, cos 0 5 1, the maximum value, and y 5 3 cos 0 5 3(1) 5 3.
When x 5 p, cos p 5 21, the minimum value, and y 5 3 cos p 5 3(21) 5 23.
a. The maximum value of the function is 3 and the minimum value is 23. Answer
b. The range of y 5 3 cos x is 23 y 3. Answer
c. The amplitude of the function is P
3 2 (23)
3
P 5 3. Answer
EXAMPLE 2
Describe the relationship between the graph of y 5 24 sin x and the graph of
y 5 sin x and sketch the graphs.
Solution
• The function g(x) 5 4f(x) is the function f(x) stretched by the factor 4 in
the vertical direction.
• The function 2g(x) is the function g(x) reflected in the x-axis.
Apply these rules to the sine function.
• The function y 5 24 sin x is the function y 5 sin x stretched by the factor
4 in the vertical direction and reflected in the x-axis.
The amplitude of y 5 24 sin x 5
4 2 (24)
2
5 4.
y
y 5 24 sin x
4
2
22p
2p
p
2p
x
y 5 4 sin x
Period
The function g(x) 5 f(ax) is the function f(x) stretched or compressed by a factor of a in the horizontal direction. Compare the graphs of y 5 sin x, y 5 sin 2x,
and y 5 sin 12x. Consider the maximum, zero, and minimum values of y for one
cycle of the graph of y 5 sin x.
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zero
maximum
zero
minimum
zero
y 5 sin x
x50
x5p
2x 5 0
2x 5 p
x 5 3p
2
3p
2x 5 2
x 5 2p
y 5 sin 2x
x 5 p2
2x 5 p2
x50
5 p4
x
5 p2
5 3p
4
5 p2
x5p
1
2x
y 5 sin 21x
x
1
2x
1
2x
50
x50
x
5 3p
2
x 5 3p
1
2x
5p
x 5 2p
2x 5 2p
x5p
1
2x
5 2p
x 5 4p
y
1
2p4 O
p
4
p
2
3p
4
p
5p
4
3p
2
7p
4
2p
9p
4
5p
2
11p
4
3p
13p
4
7p
2
15p
4
4p x
21
The graph shows the functions y 5 sin x ( ), y 5 sin 2x ( ), and
y 5 sin 12x ( ) in the interval 0 x 4p. The graph of y 5 sin 2x is the graph
of y 5 sin x compressed by the factor 12 in the horizontal direction. The graph of
y 5 sin 12x is the graph of y 5 sin x stretched by the factor 2 in the horizontal
direction.
• For y 5 sin x, there is one complete cycle in the interval 0 # x # 2p.
• For y 5 sin 2x, there is one complete cycle in the interval 0 # x # p.
• For y 5 sin 12x, there is one complete cycle in the interval 0 # x # 4p.
The difference between the x-coordinates of the endpoints of the interval
for one cycle of the graph is the period of the graph.
• The period of y 5 sin x is 2p.
• The period of y 5 sin 2x is p.
• The period of y 5 sin 12x is 4p.
In general:
The period of y 5 sin bx and y 5 cos bx is P 2p
b P.
EXAMPLE 3
a. Use a calculator to sketch the graph of y 5 cos 2x in the interval
0 x 2p.
b. What is the period of y 5 cos 2x?
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451
Solution a. With the calculator in radian mode:
ENTER:
Yⴝ
ENTER
p
WINDOW 0 ENTER
p
2 2nd
ⴜ
ENTER
DISPLAY:
2 X,T,⍜,n
COS
ENTER
2nd
6 ENTER 21.5
1.5 ENTER
GRAPH
b. The graph shows that there are two cycles of the function in the 2p interval.
The period of the graph is 2p
2 5 p. Answer
Phase Shift
The graph of f(x 1 c) is the graph of f(x) moved c units to the right when c is
negative or c units to the left when c is positive. The horizontal translation of a
trigonometric function is called a phase shift. Compare the graph of y 5 sin x
and the graph of y 5 sin A x 1 p2 B .
x
0
p
2
p
3p
2
2p
5p
2
3p
7p
2
4p
sin x
0
1
0
21
0
1
0
21
0
x 1p
2
sin A x 1 p
2B
p
2
p
3p
2
2p
5p
2
3p
7p
2
4p
9p
2
1
0
21
0
1
0
21
0
1
y
1
2p2
O
21
y 5 sin x
p
2
3p
2
p
(
y 5 sin x 1 p2
2p
5p
2
3p
7p
2
4p
x
)
The graph of y 5 sin A x 1 p2 B is the graph of y 5 sin x moved p2 units to
the left.
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EXAMPLE 4
Sketch the graph of y 5 3 sin 2 A x 1 p6 B .
Solution
How to Proceed
(1) The phase shift is 2p6 . Locate a point on
the x-axis that is p6 to the left of
A 2p6 ,
0 B , is the
the origin. This point,
starting point of one cycle of the sine curve:
(2) The period is 2p
2 5 p. Locate a point p
units to the right of the point in
y
1
2p6 O
21
p
2
5p
6
p
x
step 1. This point, A 5p
6 , 0 B , is the upper
endpoint of one cycle of the curve:
y
(3) Divide the interval for one cycle into four
parts of equal length along the x-axis:
1
5p
6
2p6
21
(4) The amplitude of the curve is 3:
(5) The y-coordinates of the sine curve
increase from 0 to the maximum in the
first quarter of the cycle, decrease from
the maximum to 0 in the second quarter,
decrease from 0 to the minimum in the
third quarter, and increase from the
minimum to 0 in the fourth quarter. Sketch
the curve:
3
x
y
2
1
p
6
21
5p
6
x
22
23
SUMMARY
For the graphs of y 5 a sin b(x 1 c) and y 5 a cos b(x 1 c):
1. The amplitude is a.
• The maximum value of the function is a and the minimum value is 2|a|.
2. The period of the function is P 2p
b P.
• There are b cycles in a 2p interval.
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453
3. The phase shift is 2c.
• The graph of y 5 sin (x 1 c) or y 5 cos (x 1 c) is the graph of y 5 sin x
or y 5 cos x shifted c units in the horizontal direction.
If c is positive, the graph is shifted |c| units to the left.
If c is negative, the graph is shifted |c| units to the right.
4. The graph of y 5 2a sin b(x 1 c) or y 5 2a cos b(x 1 c) is a reflection in
the x-axis of y 5 a sin b(x 1 c) or y 5 a cos b(x 1 c).
5. The domain of the function is the set of real numbers.
6. The range of the function is the interval [2a, a].
Exercises
Writing About Mathematics
1. Is the graph of y 5 sin 2 A x 1 p2 B the same as the graph of y 5 sin 2 A x 2 p2 B ? Justify your
answer.
2. Is the graph of y 5 cos 2 A x 1 p4 B the same as the graph of y 5 cos A 2x 1 p4 B ? Justify your
answer.
Developing Skills
In 3–10, find the amplitude of each function.
3. y 5 sin x
4. y 5 2 cos x
5. y 5 5 cos x
7. y 5 34 sin x
8. y 5 21 cos x
9. y 5 0.6 cos x
6. y 5 3 sin x
10. y 5 18 sin x
In 11–18, find the period of each function.
11. y 5 sin x
12. y 5 cos x
13. y 5 cos 3x
14. y 5 sin 2x
15. y 5 cos 12x
16. y 5 sin 13x
17. y 5 sin 1.5x
18. y 5 cos 0.75x
In 19–26, find the phase shift of each function.
19. y 5 cos A x 1 p2 B
22. y 5 sin A x 2 p4 B
25. y 5 sin 2(x 1 p)
20. y 5 cos A x 2 p2 B
23. y 5 cos A x 2 p6 B
26. y 5 cos (2x 2 p)
21. y 5 sin A x 1 p3 B
24. y 5 sin 2 A x 1 3p
4 B
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In 27–38, sketch one cycle of each function.
27. y 5 sin x
28. y 5 cos x
29. y 5 sin 2x
30. y 5 sin 12x
31. y 5 cos 3x
32. y 5 3 cos x
33. y 5 4 sin 3x
34. y 5 12 cos 31x
35. y 5 2sin 2x
36. y 5 2cos 12x
37. y 5 sin A x 1 p2 B
38. y 5 12 cos A x 2 p4 B
Applying Skills
39. Show that the graph of y 5 sin x is the graph of y 5 cos A x 2 p2 B .
40. Electromagnetic radiation emitted by a radio signal can be described by the formula
e 5 0.014 cos (2pft)
where e is in volts, the frequency f is in kilohertz (kHz), and t is time.
a. Graph two cycles of e for f 5 10 kHz.
b. What is the voltage when t 5 2 seconds?
41. As stated in the Chapter Opener, sound can be thought of as vibrating air. Simple sounds
can be modeled by a function h(t) of the form
h(t) 5 sin (2pft)
where the frequency f is in kilohertz (kHz) and t is time.
a. The frequency of “middle C” is approximately 0.261 kHz. Graph two cycles of h(t) for
middle C.
b. The frequency of C3, or the C note that is one octave lower than middle C, is approximately 0.130 kHz. On the same set of axes, graph two cycles of h(t) for C3.
c. Based on the graphs from parts a and b, the periods of each function appear to be
related in what way?
42. In 2008, the temperature (in Fahrenheit) of a city can be modeled by:
f(t) 5 71.3 1 12.1 sin (0.5t 2 1.4)
where t represents the number of months that have passed since the first of the year. (For
example, t 5 4 represents the temperature at May 1.)
a. Graph f(t) in the interval [0, 11].
b. Is it reasonable to extend this model to the year 2009? Explain.
Hands-On Activity
1. Sketch the graph of y 5 2 sin x in the interval [0, 2p].
2. Sketch the graph of the image of y 5 2 sin x under the translation T0,3.
3. Write an equation of the graph drawn in step 2.
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455
4. What are the maximum and minimum values of y for the image of y 5 2 sin x under the
translation T0,3?
5. The amplitude of the image of y 5 2 sin x under the translation T0,3 is 2. How can the maximum and minimum values be used to find the amplitude?
6. Repeat steps 2 through 5 for the translation T0,24.
11-4 WRITING THE EQUATION OF A SINE OR COSINE GRAPH
Each of the graphs that we have studied in this chapter have had an equation of
the form y 5 a sin b(x 1 c) or y 5 a cos b(x 1 c). Each of these graphs has a
maximum value that is the amplitude, a, and a minimum, 2a. The positive
number b is the number of cycles in an interval of 2p, and 2p
b is the period or the
length of one cycle. For y 5 a sin b(x 1 c), a basic cycle begins at y 5 0, and for
y 5 a cos b(x 1 c), a basic cycle begins at y 5 a. The basic cycle of the graph
closest to the origin is contained in the interval 2c x 2c 1 2p
b and the phase
shift is 2c. Using these values, we can write an equation of the graph.
y 5 a sin (bx 1 c)
1. Identify the maximum and minimum values of y for the function. Find a.
a 5 maximum 22 mininum
2. Identify one basic cycle of the sine graph that begins at y 5 0, increases to
the maximum value, decreases to 0, continues to decrease to the minimum
value, and then increases to 0. Determine the x-coordinates of the endpoints of this cycle. Write in interval notation the domain of one cycle,
x0 x # x1 or [x0, x1].
3. The period or the length of one cycle is 2p
b 5 x1 2 x0. Find b using this formula.
4. The value of c is the opposite of the lower endpoint of the interval of the
basic cycle: c 5 2x0.
y 5 a cos (bx 1 c)
1. Identify the maximum and minimum values of y for the function. Find a.
a 5 maximum 22 mininum
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2. Identify one basic cycle of the graph that begins at the maximum value,
decreases to 0, continues to decrease to the minimum value, increases to 0,
and then increases to the maximum value. Find the x-coordinates of the
endpoints of this cycle. Write in interval notation the domain of one cycle,
x0 x # x1 or [x0, x1].
3. The period or the length of one cycle is 2p
b 5 x1 2 x0. Find b using this
formula.
4. The value of c is the opposite of the lower endpoint of the interval of the
basic cycle: c 5 2x0.
EXAMPLE 1
Write an equation of the graph below in the form y 5 a cos bx.
y
2
1
p O
22p
3 23
21
2p
3
p
3
p
4p
3
5p
3
2p x
22
Solution (1) The maximum y value is 2 and the minimum y value is 22.
a5
2 2 (22)
2
52
(2) There is one cycle of the curve in the interval 2p3 x p.
(3) The period is the difference between the endpoints of the interval for one
cycle. The period is p 2 A 2p3 B or 4p
3 . Therefore:
2p
b
5 4p
3
4pb 5 6p
(4) The value of c is 2 A 2p3 B or p3 .
The phase shift is 2p3 .
b 5 32
The equation of the curve is y 5 2 cos 32 A x 1 p3 B . Answer
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457
EXAMPLE 2
Write the equation of the graph from Example 1 as a sine function.
Solution (1) The maximum y value is 2 and the minimum y value is 22.
a5
2 2 (22)
2
52
2p
(2) There is one cycle of the curve in the interval 22p
3 x 3.
(3) The period is the difference between the endpoints of the interval for one
2p
4p
cycle. The period is 2p
3 2 A 2 3 B or 3 . Therefore:
2p
b
5 4p
3
4pb 5 6p
2p
(4) The value of c is 2 A 22p
3 B or 3 .
b 5 32
The phase shift is 22p
3.
The equation of the curve is y 5 2 sin 32 A x 1 2p
3 B . Answer
Note: The equations of the sine function and the cosine function differ only in
the phase shift.
To determine the phase shift, we usually choose the basic cycle with its
lower endpoint closest to zero. For this curve, the interval for the sine function,
2p
2p
S 3 , 2p T , with the lower endpoint 3 could also have been chosen. The equation
could also have been written as y 5 2 sin 32 A x 2 2p
3 B.
Exercises
Writing About Mathematics
1. Tyler said that one cycle of a cosine curve has a maximum value at A p4 , 5 B and a minimum
p
value at A 5p
4 , 25 B . The equation of the curve is y 5 5 cos A 2x 2 2 B . Do you agree with
Tyler? Explain why or why not.
2. Is the graph of y 5 sin 2(x 1 p) the same as the graph of y 5 sin 2x? Explain why or why
not.
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Developing Skills
In 3–14, for each of the following, write the equation of the graph as: a. a sine function b. a cosine
function. In each case, choose the function with the smallest absolute value of the phase shift.
3.
4.
y
1
1
O
p
2
p
1
5.
y
3p
2
O
2p x
p
2
2p x
3p
2
p
1
y
2
6.
y
3
2
1
1
O
p
2
p
3p
2
O
2p x
1
1
p
2p
3p
2
x
2
2
7.
p
2
3
8.
y
1
y
2
1
O
2p x
p
1
O
p
2
p
2p x
3p
2
1
2
9.
10.
y
1
y
2
1
O
1
p
2
p
3p
2
2p x
O
p3
1
2
p
3
2p
3
p
4p
3
5p
3
2p x
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11.
12.
y
1
459
y
3
2
O
p
2
3p
2
p
x
1
1
O
1
p
4
p 3p
2 4
p
5p 3p 7p
4 2 4
2p
x
9p
4
2
3
13.
14.
y
1
x
O
p
2
p
3p
2
1
2p
y
2
1
x
O
p
2
1
p
3p
2
2p
2
Applying Skills
15. Motion that can be described by a sine or cosine function is called simple harmonic motion.
During the day, a buoy in the ocean oscillates in simple harmonic motion. The frequency of
the oscillation is equal to the reciprocal of the period. The distance between its high point
and its low point is 1.5 meters. It takes the buoy 5 seconds to move between its low point
and its high point, or 10 seconds for one complete oscillation from high point to high point.
Let h(t) represent the height of the buoy as a function of time t.
y
High
point
1.5 m
Low
point
x
a. What is the amplitude of h(t)?
b. What is the period of h(t)?
c. What is the frequency of h(t)?
d. If h(0) represents the maximum height of the buoy, write an expression for h(t).
e. Is there a value of t for which h(t) 5 1.5 meters? Explain.
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11-5 GRAPH OF THE TANGENT FUNCTION
We can use the table shown below to draw the graph of y 5 tan x. The values of
x are given at intervals of p6 from 22p to 2p. The values of tan x are the approximate decimal values displayed by a calculator, rounded to two decimal places.
No value is listed for those values of x for which tan x is undefined.
x
22p 211p
6
25p
3
23p
2
tan x
0
0.58
1.73
—
x
0
p
6
p
3
p
2
tan x
0
0.58
1.73
—
24p
3
27p
6
2p
25p
6
22p
3
2p2
0
0.58
1.73
—
p
7p
6
4p
3
3p
2
0
0.58
1.73
—
21.73 20.58
2p
3
5p
6
21.73 20.58
2p3
2p6
21.73 20.58
5p
3
11p
6
21.73 20.58
2p
0
y
1
O
22p
2 3p
2
2p
p
2
2p2
p
3p
2
2p x
–1
y 5 tan x
The graph of the tangent function is a curve that increases through negative
values of tan x to 0 and then continues to increase through positive values. At
odd multiples of p2 , the graph is discontinuous and then repeats the same pattern. Since there is one complete cycle of the curve in the interval from x 5 2p2
to x 5 p2 , the period of the curve is p2 2 A 2p2 B 5 p. The curve is its own image
under the transformation Tp,0.
The graph shows a vertical line at x 5 p2 and at every value of x that is an
odd multiple of p2 . These lines are vertical asymptotes. As x approaches p2 or any
odd multiple of p2 from the left, y increases; that is, y approaches infinity. As x
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Graph of the Tangent Function
461
approaches p2 or any odd multiple of p2 from the right, y decreases; that is,
y approaches negative infinity.
Compare the graph shown above with the graph displayed on a graphing
calculator.
ENTER:
Yⴝ
TAN
X,T,⍜,n
WINDOW 22
ENTER
ⴜ
DISPLAY:
p
2nd
p
2 2nd
p
2nd
ENTER
ENTER
6 ENTER
210 ENTER 10 ENTER
GRAPH
When in connected mode, the calculator displays a line connecting the
points on each side of the values of x for which tan x is undefined. The lines that
p p
3p
appear to be vertical lines at 23p
2 , 22 , 2 , and 2 are not part of the graph. On
the table given by the graphing calculator, the y-values associated with these
x-values are given as ERROR.
ENTER:
TBLSET 23
2nd
2nd
p
ⴜ
2 ENTER
2nd
p
ⴜ
6 ENTER
2nd
DISPLAY:
TABLE
X
-4.712
-4.189
-3.665
-3.142
-2.618
-2.094
-1.571
Y1
ERROR
-1.732
-.5774
0
.57735
1.7321
ERROR
X=-4.71238898038
EXAMPLE 1
Sketch one cycle of the graph of y 5 tan A x 2 p4 B .
Solution The graph of y 5 tan A x 2 p4 B is the graph of
y 5 tan x with a phase shift of p4 . Since there is
one cycle of y 5 tan x in the interval 2p2 x p2 ,
there will be one cycle of y 5 tan A x 1 p4 B in the
interval 2p2 1 p4 x p2 1 p4 , that is, in the
interval 2p4 x 3p
4.
y
2p4
p p 3p
4 2 4
px
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SUMMARY
For the graph of y 5 tan x:
1. There is no amplitude.
• The graph has no maximum or minimum values.
2. The period of the function is p.
3. The domain of the function is U x : x p2 1 np for n an integer V .
• For integral values of n, the graph has vertical asymptotes at x 5 p2 1 np.
4. The range of the function is {Real numbers} or (2`, `).
Exercises
Writing About Mathematics
1. List at least three ways in which the graph of the tangent function differs from the graph of
the sine function and the cosine function.
2. Does y 5 tan x have a maximum and a minimum value? Justify your answer.
Developing Skills
3p
3. Sketch the graph of y 5 tan x from x 5 23p
2 to x 5 2 .
a. What is the period of y 5 tan x?
b. What is the domain of y 5 tan x?
c. What is the range of y 5 tan x?
4. a. Sketch the graph of y 5 tan x from x 5 0 to x 5 2p.
b. On the same set of axes, sketch the graph of y 5 cos x from x 5 0 to x 5 2p.
c. For how many pairs of values does tan x 5 cos x in the interval [0, 2p]?
5. a. Sketch the graph of y 5 tan x from x 5 2p2 to x 5 p2 .
b. Sketch the graph of y 5 tan (2x) from x 5 2p2 to x 5 p2 .
c. Sketch the graph of y 5 2tan x from x 5 2p2 to x 5 p2 .
d. How does the graph of y 5 tan (2x) compare with the graph of y 5 2tan x?
Applying Skills
6. The volume of a cone is given by the formula
h
V 5 13Bh
where B is the area of the base and h is the height of the cone.
u
a. Find a formula for h in terms of r and u when r is the radius of the base
and u is the measure of the angle that the side of the cone makes with the base.
b. Use part a to write a formula for the volume of the cone in terms of r and u.
r
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463
7. Recall from your geometry course that a polygon is circumscribed about a circle if each side
of the polygon is tangent to the circle. Since each side is tangent to the circle, the radius of
the circle is perpendicular to each side at the point of tangency. We will use the tangent
function to examine the formula for the perimeter of a circumscribed regular polygon.
a. Let square ABCD be circumscribed about circle O. A radius
of the circle, OP, is perpendicular to AB at P.
s
D
C
(1) In radians, what is the measure of AOB?
(2) Let mAOP 5 u. If u is equal to one-half the measure of
AOB, find u.
s
O
s
r
(3) Write an expression for AP in terms of tan u and r, the
radius of the circle.
(4) Write an expression for AB 5 s in terms of tan u and r.
b. Let regular pentagon ABCDE be circumscribed
about circle O. Repeat part a using pentagon
ABCDE.
c. Do you see a pattern in the formulas for the perimeter of the square and of the pentagon? If so, make a
conjecture for the formula for the perimeter of a circumscribed regular polygon in terms of the radius r
and the number of sides n.
P
A
(5) Use part (4) to write an expression for the perimeter in
terms of r and the number of sides, n.
B
D
s
s
E
C
O
s
s
r
A
P
B
11-6 GRAPHS OF THE RECIPROCAL FUNCTIONS
The Cosecant Function
The cosecant function is defined in terms of the sine function: csc x 5 sin1 x. To
graph the cosecant function, we can use the reciprocals of the sine function values. The reciprocal of 1 is 1 and the reciprocal of a positive number less than 1
is a number greater than 1. The reciprocal of 21 is 21 and the reciprocal of a
negative number greater than 21 is a number less than 21. The reciprocal of 0
is undefined. Reciprocal values of the sine function exist for 21 sin x 0, and
for 0 sin x 1. Therefore:
2` csc x 21
1 csc x `
(Recall that the symbol ` is called infinity and is used to indicate that a set
of numbers has no upper bound. The symbol 2` indicates that the set of
Page 464
Graphs of Trigonometric Functions
numbers has no lower bound.) For values of x that are multiples of p, sin x 5 0
and csc x is undefined. For integral values of n, the vertical lines on the graph at
x 5 np are asymptotes.
csc x
y
y5
y 5 sin x
1
2p2
p
2
O
3p
2
p
–1
2p x
csc x
2p
y5
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The Secant Function
The secant function is defined in terms of the cosine function: sec x 5 cos1 x. To
graph the secant function, we can use the reciprocals of the cosine function
values. Reciprocal values of the cosine function exist for 21 cos x 0, and for
0 cos x 1. Therefore:
2` sec x 21
1 sec x `
For values of x that are odd multiples of p2 , cos x 5 0 and sec x is undefined. For integral values of n, the vertical lines on the graph at x 5 p2 1 np are
asymptotes.
y
x
x
sec
sec
y5
y5
y 5 cos x
1
2p2
2p
O
p
2
3p
2
p
–1
2p x
y5
y5
sec
sec
x
x
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465
The Cotangent Function
y 5 tan
x
tx
y 5 tan
x
y 5 co
tx
tx
y 5 co
y 5 co
y
x
The cotangent function is defined in terms of the tangent function: cot x 5 tan1 x.
To graph the cotangent function, we can to use the reciprocals of the tangent
function values. For values of x that are multiples of p, tan x 5 0 and cot x is
undefined. For values of x for which tan x is undefined, cot x 5 0. For integral
values of n, the vertical lines on the graph at x 5 np are asymptotes.
y 5 tan
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1
2p
2p2
O
–1
p
2
p
3p
2
2p x
EXAMPLE 1
Use a calculator to sketch the graphs of y 5 sin x and y 5 csc x for
22p x 2p.
Solution There is no key for the cosecant function on the TI graphing calculator. Therefore, the function must be entered in terms of the sine
function.
ENTER:
Yⴝ
SIN
ENTER
X,T,⍜,n
X,T,⍜,n
Yⴝ 1
)
ⴜ
GRAPH
)
SIN
DISPLAY:
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SUMMARY
y 5 csc x
y 5 sec x
y 5 cot x
Amplitude:
none
none
none
Maximum:
1`
1`
1`
Minimum:
2`
2`
2`
Period:
2p
2p
p
: x p2 1 np V
(2`, 21] < [1, `)
{x : x np}
Ux
{x : x np}
Domain:
(2`, 21] < [1, `)
Range:
(2`, `)
Exercises
Writing About Mathematics
1. If tan x increases for all values of x for which it is defined, explain why cot x decreases for
all values of x for which it is defined.
2. In the interval 0 x p, cos x decreases. Describe the change in sec x in the same interval.
Developing Skills
In 3–10, match each graph with its function.
(1)
(2)
p
2p
x
(5)
(3)
y
y
2p
p
x
(6)
(4)
y
y
p
2p
x
2p
p
x
y
y
x
x
p
p
2p
(8)
(7)
y
y
2p
x
2p
3. y 5 csc x
4. y 5 sec x
5. y 5 cot x
7. y 5 csc x2
8. y 5 sec 2x
9. y 5 2cot x
p
2p
6. y 5 2sec x
10. y 5 cot A x 1 p2 B
11. a. Sketch the graphs of y 5 sin x and y 5 csc x for 22p x 2p.
b. Name four values of x in the interval 22p x 2p for which sin x 5 csc x.
p
x
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467
12. a. Sketch the graphs of y 5 cos x and y 5 sec x for 22p x 2p.
b. Name four values of x in the interval 22p x 2p for which cos x 5 sec x.
13. a. Sketch the graphs of y 5 tan x and y 5 cot x for 2p x p.
b. Name four values of x in the interval 2p x p for which tan x 5 cot x.
14. List two values of x in the interval 22p x 2p for which sec x is undefined.
15. List two values of x in the interval 22p x 2p for which csc x is undefined.
16. List two values of x in the interval 22p x 2p for which tan x is undefined.
17. List two values of x in the interval 22p x 2p for which cot x is undefined.
18. The graphs of which two trigonometric functions have an asymptote at x 5 0?
19. The graphs of which two trigonometric functions have an asymptote at x 5 p2 ?
20. Using the graphs of each function, determine whether each function is even, odd, or
neither.
a. y 5 tan x
b. y 5 csc x
c. y 5 sec x
d. y 5 cot x
Applying Skills
21. A rotating strobe light casts its light on the ceiling of the
community center as shown in the figure. The light is
located 10 feet from the ceiling.
a. Express a, the distance from the light to its projection
on the ceiling, as a function of u and a reciprocal
trigonometric function.
10 ft
a
u
b. Complete the following table, listing each value to the
nearest tenth.
u
p
18
p
9
p
6
2p
9
a
c. The given values of u in part b increase in equal increments. For these values of u, do
the distances from the light to its projection on the ceiling increase in equal increments? Explain.
d. The maximum value of u, in radians, before the light stops shining on the ceiling is 4p
9.
To the nearest tenth, how wide is the ceiling?
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11-7 GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS
The trigonometric functions are not one-to-one functions. By restricting the
domain of each function, we were able to write inverse functions in Chapter 10.
Inverse of the Sine Function
The graph of y 5 sin x is shown below on the left. When the graph of y 5 sin x
is reflected in the line y = x, the image, x 5 sin y or y 5 arcsin x, is a relation that
is not a function. The interval 2p2 x p2 includes all of the values of sin x from
21 to 1.
y
y
p
2
y5x
( 12, p6)
1
22p
–1 O1
–1
2p x
y 5 sin x
x 5 sin y
(!23, p3)
(!22, p4)
O
1
21
(2!2 2 , 2p4 )
(2!2 3 , 2p3 )
(
)
21, 2p
2 6
x
y 5 arcsin x
2p
2
If we restrict the domain of the sine function to 2p2 x p2 , that subset of
the sine function is a one-to-one function and has an inverse function. When we
reflect that subset over the line y = x, the image is the function y 5 arcsin x or
y 5 sin–1 x.
Sine Function with
a Restricted Domain
Inverse Sine Function
y 5 sin x
y 5 arcsin x
Domain 5 U x : 2p2 # x # p2 V
Range 5 {y : 21 y 1}
Domain 5 {x : 21 x 1}
Range 5 U y : 2p2 # y # p2 V
Inverse of the Cosine Function
The graph of y 5 cos x is shown on the top of page 469. When the graph of
y 5 cos x is reflected in the line y = x, the image, x 5 cos y or y 5 arccos x, is a
relation that is not a function. The interval 0 x p includes all of the values
of cos x from 21 to 1.
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Graphs of Inverse Trigonometric Functions
y
y
(2!2 3 , 5p6 )
(2!2 2 , 3p4 )
p
y5x
p
(212, 2p3 )
2p
p
x
y 5 cos x
O
(0, p2)
( 12, p3)
(!22, p4)
(!23, p6)
2p
y 5 arccos x
x 5 cos y
O
21
x
1
If we restrict the domain of the cosine function to 0 x p, that subset of
the cosine function is a one-to-one function and has an inverse function. When
we reflect that subset over the line y = x, the image is the function y 5 arccos x
or y 5 cos21 x.
Cosine Function with a
Restricted Domain
Inverse Cosine Function
y 5 cos x
y 5 arccos x
Domain 5 {x : 0 x p}
Domain 5 {x : 21 x 1}
Range 5 {y : 21 y 1}
Range 5 {y : 0 y p}
Inverse of the Tangent Function
The graph of y 5 tan x is shown below on the left. When the graph of y 5 tan y
is reflected in the line y 5 x, the image, x 5 tan y or y 5 arctan x, is a relation
that is not a function. The interval 2p2 x p2 includes all of the values of tan
x from 2` to `.
y
3p
2
x5tan y
p
p
2
2 3p
2
2p
2p
2
2p
2
x5tan y
p
2
y5tan x
2 3p
2
p
3p
2
p
2
!3,
(1, p4 ) (
(!33 , p6)
x
x5tan y
2p
y5tan x
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(21, 2p4 ) (2!3 , 2p)
3
6
(2!3, 2p3 )
y 5 arctan x
p
22
p
3
)
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Graphs of Trigonometric Functions
If we restrict the domain of the tangent function to 2p2 x p2 , that subset of the tangent function is a one-to-one function and has an inverse function. When we reflect that subset over the line y = x, the image is the function
y 5 arctan x or y 5 tan21 x.
Tangent Function with a
Restricted Domain
Inverse Tangent Function
y 5 tan x
y 5 arctan x
Domain 5 U x : 2p2 , x , p2 V
Domain 5 {x : x is a real number}
Range 5 {y : y is a real number}
Range 5 U y : 2p2 , y , p2 V
If we know that sin x 5 0.5738, a calculator will give the value of x in the
restricted domain. We can write x 5 arcsin 0.5770. On the calculator, the
SINⴚ1 key is used for the arcsin function.
To write the value of x in degrees, change the calculator to degree mode.
ENTER:
MODE
ENTER
CLEAR
SINⴚ1 0.5738
)
DISPLAY: s i n - 1 ( 0 . 5 7 3 8 )
35.01563871
2nd
ENTER
In degrees, sin 35° 0.5738. This is the value of x for the restricted domain
of the sine function. There is also a second quadrant angle whose reference
angle is 35°. That angle is 180° 2 35° or 145°. These two measures, 35° and 145°,
and any measures that differ from one of these by a complete rotation, 360°, are
a value of x. Therefore, if sin x 5 0.5738,
x 5 35 1 360n or x 5 145 1 360n
for all integral values of n.
EXAMPLE 1
Find, in radians, all values of u such that tan u 5 21.200 in the interval
0 u 2p. Express the answer to four decimal places.
Solution If tan u 5 21.200, u 5 arctan 2 1.200.
Set the calculator to radian mode. Then use the TANⴚ1 key.
ENTER:
2nd
ENTER
TANⴚ1 21.200
)
DISPLAY:
tan-1(-1.200)
-.8760580506
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Graphs of Inverse Trigonometric Functions
To four decimal places, u 5 20.8761. This is the
value of u in the restricted domain of the tangent
function. An angle of 20.8761 radians is a fourthquadrant angle. The fourth-quadrant angle with the
same terminal side and a measure between 0 and
2p is 20.8761 1 2p 5.4070. The tangent function
values are negative in the second and fourth quadrants. There is a second-quadrant angle with the
same tangent value. The first-quadrant reference
angle has a measure of 0.8761 radians. The second
quadrant angle with a reference angle of 0.8761
radians has a measure of p 2 0.8761 2.2655.
y
u
R(1, 0)
x
21.2
Answer x 2.2655 and x 5.4070
Exercises
Writing About Mathematics
1. Show that if arcsin 212 5 x, then the measure of the reference angle for x is 30°.
2. Is arctan 1 5 220° a true statement? Justify your answer.
Developing Skills
In 3–14, find each exact value in degrees.
3. y 5 arcsin 12
6. y 5 arctan !3
9. y 5 arcsin
12. y 5 arcsin
A 2 !3
2 B
A 2 !2
2 B
4. y 5 arccos 12
5. y 5 arctan 1
7. y 5 arcsin (21)
8. y 5 arccos 0
10. y 5 arccos
A 2 !2
2 B
13. y 5 arctan 0
11. y 5 arctan (21)
14. y 5 arccos (21)
In 15–26, find each exact value in radians, expressing each answer in terms of p.
15. y 5 arcsin 1
18. y 5 arcsin A !3
2 B
16. y 5 arccos 1
19. y 5 arcsin A 2 !3
2 B
17. y 5 arctan 1
20. y 5 arccos 12
21. y 5 arccos A 212 B
22. y 5 arctan !3
23. y 5 arctan A 2!3 B
24. y 5 arctan 0
25. y 5 arccos 0
26. y 5 arcsin 0
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In 27–32, for each of the given inverse trigonometric function values, find the exact function value.
27. sin (arccos 1)
28. cos (arcsin 1)
30. sin A arccos 2 !3
2 B
29. tan (arctan 1)
32. cos A arccos 212 B
31. sin (arctan 21)
33. a. On the same set of axes, sketch the graph of y 5 arcsin x and of its inverse function.
b. What are the domain and range of each of the functions graphed in part a?
34. a. On the same set of axes, sketch the graph of y 5 arccos x and of its inverse function.
b. What are the domain and range of each of the functions graphed in part a?
35. a. On the same set of axes, sketch the graph of y 5 arctan x and of its inverse function.
b. What are the domain and range of each of the functions graphed in part a?
Applying Skills
Finish
line
36. A television camera 100 meters from the
starting line is filming a car race, as shown in
the figure. The camera will follow car number 2.
a. Express u as a function of d, the distance
of the car to the starting line.
2
300 m
4
b. Find u when the car is 50 meters from the
starting line.
c. If the finish line is 300 meters away from
the starting line, what is the maximum
value of u to the nearest minute?
d
u
Starting
line
100 m
11-8 SKETCHING TRIGONOMETRIC GRAPHS
One cycle of the basic sine curve and of
the basic cosine curve are shown to the
right. In the previous sections, we have
seen how the values of a, b, and c
change these curves without changing
the fundamental shape of a cycle of
the graph. For y 5 a sin b(x 1 c) and for
y 5 a cos b(x 1 c):
p
2p
y = sin x
• a 5 amplitude
• b 5 number of cycles in a 2p
interval
•
2p
b
5 period of the graph
• 2c 5 phase shift
p
y = cos x
2p
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Sketching Trigonometric Graphs
473
EXAMPLE 1
a. Sketch two cycles of the graph of y 5 2 sin A x 2 p4 B without using a
calculator.
b. On the same set of axes, sketch one cycle of the graph of y 5 cos 12x without
using a calculator.
c. In the interval 0 x 4p, how many points do the two curves have in
common?
Solution
y
2
1
O
21
p
4
p
2
3p
4
p
5p 3p 7p
2
4
4
2p
9p 5p 11p
4
2
4
3p 13p
4
7p 15p
2
4
4p
17p
4
x
22
a. For the function y 5 2 sin A x 2 p4 B , a 5 2, b 5 1, and c 5 2p4 . Therefore,
one cycle begins at x 5 p4 . There is one complete cycle in the 2p interval,
that is from p4 to 9p
4 . Divide this interval into four equal intervals and sketch
one cycle of the sine curve with a maximum of 2 and a minimum of 22.
17p
There will be a second cycle in the interval from 9p
4 to 4 . Divide this
interval into four equal intervals and sketch one cycle of the sine curve with
a maximum of 2 and a minimum of 22.
y
2
(
y 5 2 sin x 2 p4
)
1
O
21
p
3p
2p
y 5 cos
22
4p
x
( 12 x)
b. For the function y 5 cos 21x, a 5 1, b 5 12, and c 5 0. Therefore, one cycle
begins at the origin, x 5 0. There is one-half of a complete cycle in the 2p
interval, so the interval for one cycle is 2p
5 4p. Divide the interval from
1
2
0 to 4p into four equal intervals and sketch one cycle of the cosine curve
with a maximum of 1 and a minimum of 21.
c. The curves have four points in common. Answer
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Exercises
Writing About Mathematics
1. Calvin said that the graph of y 5 tan A x 2 p4 B has asymptotes at x 5 3p
4 1 np for all integral values of n. Do you agree with Calvin? Explain why or why not.
2. Is the graph of y 5 sin A 2x 2 p4 B the graph of y 5 sin 2x moved p4 units to the right?
Explain why or why not.
Developing Skills
In 3–14, sketch one cycle of the graph.
3. y 5 2 sin x
4. y 5 3 sin 2x
8. y 5 3 sin A x 2 p3 B
12. y 5 tan A x 2 p2 B
7. y 5 4 cos 2x
11. y 5 tan x
6. y 5 2 sin 12x
5. y 5 cos 3x
9. y 5 cos 2 A x 1 p6 B
13. y 5 22 sin x
10. y 5 4 sin (x 2 p)
14. y 5 2cos x
In 15–20, for each of the following, write the equation of the graph as: a. a sine function b. a cosine
function. In each case, choose the function with the smallest absolute value of the phase shift.
15.
y
16.
y
1
1
p
2
p
2
2 3p
2 2p 2
p
3p
2
y
17.
19.
p
2p 2p2
2 3p
2
3p
2
p
2
p
3p
2
x
p
2
p
3p
2
x
1
1
2p 2p2
2 3p
2
3p
2
y
20.
y
p
1
x
p
2
p
2
y
18.
1
2p 2p
2 3p
2
2
2p 2p2
2 3p
2
x
x
p
2
p
3p
2
x
2p2
2 3p
2 2p
21. a. On the same set of axes, sketch the graphs of y 5 2 sin x and y 5 cos x in the interval
0 x 2p.
b. How many points do the graphs of y 5 2 sin x and y 5 cos x have in common in the
interval 0 x 2p?
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22. a. On the same set of axes, sketch the graphs of y 5 tan x and y 5 cos A x 1 p2 B in the interval 2p2 x 3p
2.
b. How many points do the graphs of y 5 tan x and y 5 cos A x 1 p2 B have in common in
the interval 2p2 x 3p
2?
23. a. On the same set of axes, sketch the graphs of y 5 sin 3x and y 5 2 cos 2x in the interval
0 x 2p.
b. How many points do the graphs of y 5 sin 3x and y 5 2 cos 2x have in common in the
interval 0 x 2p?
CHAPTER SUMMARY
Domain
(n an integer)
Range
Period
y 5 sin x
All real numbers
[21, 1]
2p
increase
decrease
decrease
increase
y 5 cos x
All real numbers
[21, 1]
2p
decrease
decrease
increase
increase
y 5 tan x
x p2 1 np
x np
All real numbers
p
increase
increase
increase
increase
(2`, 21] < [1, `)
2p
decrease
increase
increase
decrease
(2`, 21] < [1, `)
2p
increase
increase
decrease
decrease
All real numbers
p
decrease
decrease
decrease
decrease
y 5 csc x
y 5 sec x
y 5 cot x
x p2 1 np
x np
A 0, p2 B
A p2 , p B
A p, 3p
2 B
A 3p
2 , 2p B
For y 5 a sin b(x 1 c) or y 5 a cos b(x 1 c):
• The amplitude, a, is the maximum value of the function, and 2a is the
minimum value.
• The frequency b, is the number of cycles in the 2p interval, and the
period, 2p
, is the length of the interval for one cycle.
b
• The phase shift is 2c. If c is positive, the graph is shifted c units to the
left. If c is negative, the graph is shifted c units to the right and the fre|b|
quency, 2p
, is the reciprocal of the period.
The graphs of the trigonometric functions are periodic curves. Each graph
of y 5 a sin b(x 1 c) or of y 5 a cos b(x 1 c) is its own image under the translation (x, y) → A x, y 1 2p
. The graph of y 5 tan x is its own image under the
b B
translation (x, y) → (x, y 1 p).
When the domain of a trigonometric function is restricted to a subset for
which the function is one-to-one, the function has an inverse function.
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Function
Restricted Domain
Inverse Function
y 5 sin x
2p2 x p2
y 5 arcsin x or y 5 sin21 x
y 5 cos x
0xp
y 5 arccos x or y 5 cos21 x
y 5 tan x
2p2 x p2
y 5 arctan x or y 5 tan21 x
y
p
y
p
2
y
p
2
y 5 arcsin x
O
21
x
1
y 5 arccos x
21
p
22
2 p2
21 O
O 1
x
y 5 arctan x
x
1
VOCABULARY
11-1 Cycle • Periodic function • Period • Odd function
11-2 Even function
11-3 Amplitude • Phase shift
11-4 Simple harmonic motion • Frequency
REVIEW EXERISES
In 1–6, for each function, state: a. the amplitude b. the period c. the frequency
d. the domain e. the range. f. Sketch one cycle of the graph.
1. y 5 2 sin 3x
4. y 5 cos 2 A x 2 p3 B
2. y 5 3 cos 12x
3. y 5 tan x
5. y 5 sin (x 1 p)
6. y 5 22 cos x
In 7–10, for each graph, write the equation in the form: a. y 5 a sin b(x 1 c)
b. y 5 a cos b(x 1 c). In each case, choose one cycle with its lower endpoint closest to zero to find the phase shift.
7.
2
1
2p2
21
22
8.
y
2
1
x
O
p
2
p
3p
2
2p
2p 2p2
O
21
22
y
p
2
p
x
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Review Exercises
9.
3
2
1
10.
y
y
1
x
2p 2p O
2
21
x
2p2 O
21
22
23
p
2
p
3p
2
2p
p
2
p
3p
2
In 11–14, match each graph with its function.
(1)
(2)
y
y
x
x
p
p
2
2
2 3p
2 2p 2
(3)
p
3p
2
2p
2 3p
2 2p 2
2
(4)
y
p
2
p
3p
2
p
3p
2
2
y
x
2p 2p2
2 3p
2
11. y 5 sec x2
13. y 5 2cot A x 1 p2 B
p
2
p
3p
2
2
x
p
2p 2 2
2 3p
2
p
2
2
12. y 5 csc A x 1 p2 B
14. y 5 2tan A x 2 p2 B
In 15–20, find, in radians, the exact value of y for each trigonometric function.
15. y 5 arcsin 21
16. y 5 sin–1 1
17. y 5 arctan !3
3
20. y 5 arctan A 2 !3
18. y 5 arccos A 2 !3
19. y 5 cos–1 !2
2 B
2
3 B
21. a. What is the restricted domain for which y 5 sin x is a one-to-one
function?
b. What is the domain of the function y 5 arcsin x?
c. What is the range of the function y 5 arcsin x?
d. Sketch the graph of the function y 5 arcsin x.
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22. a. Sketch the graph of y 5 cos x in the interval 22p x 2p.
b. On the same set of axes, sketch the graph of y 5 csc x.
c. How many points do y 5 cos x and y 5 csc x have in common?
23. What are the equations of the asymptotes of the graph of y 5 tan x in the
interval 22p x 2p?
24. a. On the same set of axes, sketch the graph of y 5 2 cos x and y 5 sin 21x
in the interval 2p x p.
b. From the graph, determine the number of values of x for which
cos x 5 sin 12x in the interval 2p x p.
25. Is the domain of y 5 csc x the same as the domain of y 5 sin x? Explain
why or why not.
26. The function p(t) 5 85 1 25 sin (2pt) approximates the blood pressure of
Mr. Avocado while at rest where p(t) is in milligrams of mercury (mmHg)
and t is in seconds.
a. Graph p(t) in the interval [0, 3].
b. Find the period of p(t).
c. Find the amplitude of p(t).
d. The higher value of the blood pressure is called the systolic pressure.
Find Mr. Avocado’s systolic pressure.
e. The lower value of the blood pressure is called the diastolic pressure.
Find Mr. Avocado’s diastolic pressure.
27. The water at a fishing pier is 11 feet deep at low tide and 20 feet deep at
high tide. On a given day, low tide is at 6 A.M. and high tide is at 1 P.M. Let
h(t) represent the height of the tide as a function of time t.
a. What is the amplitude of h(t)?
y
b. What is the period of h(t)?
c. If h(0) represents the height of the
tide at 6 A.M., write an expression
for h(t).
20 ft
High tide
11 ft
Low tide
x
Exploration
Natural phenomena often occur in a cyclic pattern that can be modeled by a
sine or cosine function. For example, the time from sunrise to sunset for any
given latitude is a maximum at the beginning of summer and a minimum at the
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479
beginning of winter. If we plot this difference at weekly intervals for a year,
beginning with the first day of summer, the curve will closely resemble a cosine
curve after the translation T0,d. (A translation T0,d moves the graph of a function
d units in the vertical direction.) The equation of the cosine curve can then be
written as y 5 a cos b(x 1 c) 1 d.
STEP 1. Research in the library or on the Internet to find the time of sunrise and
sunset at weekly intervals. Let the week of June 21 be week 0 and the
week of June 21 for the next year be week 52. Round the time from
sunrise to sunset to the nearest quarter hour. For example, let 14 hours
STEP
STEP
STEP
STEP
12 minutes be 1414 hours and 8 hours 35 minutes be 812 hours.
2. Plot the data.
3. What is the amplitude that most closely approximates the data?
4. What is the period that most closely approximates the data?
5. Let d equal the average of the maximum and minimum values of the
data or:
minimum
d 5 maximum 1
2
Find an approximate value for d.
STEP 6. Write a cosine function of the form y 5 a cos b(x 1 c) 1 d that approx-
imates the data.
CUMULATIVE REVIEW
CHAPTERS 1–11
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. The sum !225 1 !29 is equal to
(1) !234
(2) 28i
(3) 8i
(4) 34i
2. The solution set of 2x 1 2 2 4 5 0 is
(1) (2) {1}
(3) {1, 21}
(4) {1, 23}
3. In radians, 225° is equivalent to
(1) p4
(2) 3p
4
(4) 7p
4
(3) 5p
4
4. Which of the following is a geometric sequence?
(1) 1, 2, 4, 7, 11, . . .
(3) 1, 1, 2, 3, 5, . . .
(2) 1, 2, 3, 4, 5, . . .
(4) 1, 2, 4, 8, 16, . . .
5. Which of the following functions is one-to-one?
(1) f(x) 5 2x2
(3) f(x) 5 2x
(2) f(x) 5 2x
(4) f(x) 5 2 tan x
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6. When written with a rational denominator, 2 23 !2 is equal to
3A2 1 !2B
(1)
(3) 3(1 1 !2)
2
(2)
3A2 2 !2B
2
(4) 2 12 !2
7. The solution set of 2x2 1 5x 2 3 5 0 is
(1) U12, 3 V
(2) U12, 23 V
(3) U212, 3 V
(4) U 212, 23 V
8. If g(x) 5 x2 and f(x) 5 2x 1 1, then g(f(x)) equals
(1) (2x 1 1)2
(3) (2x 1 1)(x2)
(2) 2x2 1 1
(4) x2 1 2x 1 1
9. If u 5 arcsin Q2 !3
2 R , then in radians, u is equal to
(1) 2p3
(2) 2p6
(3) p3
(4) 2p
3
10. The coordinates of the center of the circle (x 1 3)2 1 (y 2 2)2 5 9 are
(1) (3, 22)
(3) (23, 2)
(2) (3, 2)
(4) (23, 22)
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Solve for x and graph the solution set on the number line:
2x 2 5 7
12. Find, to the nearest degree, all values of u in the interval 0 u 360 for
which tan u 5 21.54.
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Express the roots of x3 2 8x2 1 25x 5 0 in simplest form.
5
14. Find the value of a 3(2) n21.
n50
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Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. Given: ABCDEFGH is a cube with
sides of length 1.
a. Find the exact measure of GBC,
the angle formed by diagonal BG
and side BC.
H
E
b. Find, to the nearest degree, the measure of GAC, the angle formed by
diagonals AG and AC.
16. Find the solution set of the following
system of equations algebraically.
y 5 2x2 2 3x 2 5
2x 2 y 5 7
G
F
D
C
A
B
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CHAPTER
12
CHAPTER
TABLE OF CONTENTS
12-1 Basic Identities
12-2 Proving an Identity
12-3 Cosine (A 2 B)
12-4 Cosine (A 1 B)
12-5 Sine (A 2 B) and Sine (A 1 B)
12-6 Tangent (A 2 B) and
Tangent (A 1 B)
12-7 Functions of 2A
12-8 Functions of 12A
TRIGONOMETRIC
IDENTITIES
When a busy street passes through the business
center of a town, merchants want to insure maximum
parking in order to make stopping to shop convenient.
The town planners must decide whether to allow parallel parking (parking parallel to the curb) on both sides
of the street or angle parking (parking at an angle with
the curb). The size of angle the parking space makes
with the curb determines the amount of road space
needed for parking and may limit parking to only one
side of the street. Problems such as this require the use
of function values of the parking angles and illustrate
one way trigonometric identities help us to solve
problems.
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
u
482
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483
12-1 BASIC IDENTITIES
Throughout our study of mathematics, we have used the solution of equations
to solve problems. The domain of an equation is the set of numbers for which
each side of the equation is defined. When the solution set is a proper subset of
the domain, the equation is a conditional equation. When the solution set is the
domain of the equation, the equation is an identity.
Recall that an identity is an equation that is true for all possible replacements of the variable.
In the following examples, the domain is the set of real numbers.
Algebraic Equations
Conditional equation:
x2 1 3x 2 10 5 0
Solution set: {25, 2}
Identity:
3x 1 12 5 3(x 1 4)
Solution set: {Real numbers}
Trigonometric Equations
Conditional equation:
sin u 5 1
Solution set: U p2 1 2pn, n 5 an integer V
Identity:
sin2 u 1 cos2 u 5 1
Solution set: {Real numbers}
When stated in the form of an equation, a property of the real numbers or
an application of a property of the real numbers is an algebraic identity. For
example, the additive identity property can be expressed as an algebraic identity: a 1 0 5 a is true for all real numbers.
When we defined the six trigonometric functions, we proved relationships
that are true for all values of u for which the function is defined. There are eight
basic trigonometric identities.
Pythagorean Identities
Reciprocal Identities
Quotient Identities
cos2 u 1 sin2 u 5 1
1 1 tan2 u 5 sec2 u
sec u 5 cos1 u
csc u 5 sin1 u
sin u
tan u 5 cos
u
cos u
cot u 5 sin u
cot2 u 1 1 5 csc2 u
cot u 5 tan1 u
Each of these identities is true for all values of u for which both sides of the
identity are defined. For example, cos2 u 1 sin2 u 5 1 is true for all real numbers
and 1 1 tan2 u 5 sec2 u is true for all real numbers except u 5 p2 1 np when n
is an integer.
We can use the eight basic identities to write other equations that are true
for all replacements of the variable for which the function values exist.
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Trigonometric Identities
EXAMPLE 1
Use the basic identities to show that tan u csc u 5 sec u for all values of u for
which each side of the equation is defined.
Solution Each of the functions in the given equation can be written in terms of sin u,
cos u, or both.
?
(1) Use the basic identities to write
each side of the identity in terms
of sin u and cos u:
tan u csc u 5 sec u
(2) Divide a numerator and a denominator
of the left side of the equation by sin u:
sin u
1
A cos
u B A sin u B
sin u
1
A cos
u B A sin u B
5 cos1 u
?
1
?
5 cos1 u
1
1
cos u
Note: If csc u is defined, sin u 0.
5 cos1 u ✔
EXAMPLE 2
Use the Pythagorean identities to write:
a. sin u in terms of cos u.
Solution a. cos2 u 1 sin2 u 5 1
sin2 u 5 1 2 cos2 u
sin u 5 6"1 2 cos2 u
b. cos u in terms of sin u.
b. cos2 u 1 sin2 u 5 1
cos2 u 5 1 2 sin2 u
cos u 5 6"1 2 sin2 u
Note: For a given value of u, the sign of cos u or of sin u depends on the quadrant in which the terminal side of the angle lies:
• When u is a first-quadrant angle, sin u 5 "1 2 cos2 u and
cos u 5 "1 2 sin2 u.
• When u is a second-quadrant angle, sin u 5 "1 2 cos2 u and
cos u 5 2"1 2 sin2 u.
• When u is a third-quadrant angle, sin u 5 2"1 2 cos2 u and
cos u 5 2"1 2 sin2 u.
• When u is a fourth-quadrant angle, sin u 5 2"1 2 cos2 u and
cos u 5 "1 2 sin2 u.
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Exercises
Writing About Mathematics
1. If we know the value of sin u, is it possible to find the other five trigonometric function values? If not, what other information is needed?
2. a. Explain how the identities 1 1 tan2 u 5 sec2 u and cot2 u 1 1 5 csc2 u can be derived
from the identity cos2 u 1 sin2 u 5 1.
b. The identity cos2 u 1 sin2 u 5 1 is true for all real numbers. Are the identities
1 1 tan2 u 5 sec2 u and cot2 u 1 1 5 csc2 u also true for all real numbers? Explain your
answer.
Developing Skills
In 3–14, write each expression as a single term using sin u, cos u, or both.
3. tan u
4. cot u
5. sec u
6. csc u
7. cot u sec u
8. tan2 u 1 1
9. cot2 u 1 1
u
12. tan
cot u 1 tan u cot u
10. tan u sec u cot u
11. sec u1csc u
13. tan1 u 1 cot u
14. sec u 1 csc1 u
12-2 PROVING AN IDENTITY
The eight basic identities are used to prove other identities. To prove an identity
means to show that the two sides of the equation are always equivalent. It is
generally more efficient to work with the more complicated side of the identity
and show, by using the basic identities and algebraic principles, that the two
sides are the same.
Tips for Proving an Identity
To prove an identity, use one or more of the following tips:
1. Work with the more complicated side of the equation.
2. Use basic identities to rewrite unlike functions in terms of the same
function.
3. Remove parentheses.
4. Find common denominators to add fractions.
5. Simplify complex fractions and reduce fractions to lowest terms.
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EXAMPLE 1
Prove that sec u sin u 5 tan u is an identity.
Solution Write the left side of the equation in terms of sin u and cos u.
?
sec u sin u 5 tan u
1
cos u sin
?
u 5 tan u
sin u ?
cos u 5
tan u
tan u 5 tan u ✔
Proof begins with what is known and proceeds to what is to be proved.
Although we have written the proof in Example 1 by starting with what is to be
proved and ending with what is obviously true, the proof of this identity really
begins with the obviously true statement:
tan u 5 tan u; therefore, sec u sin u 5 tan u.
EXAMPLE 2
Prove that sin u (csc u 2 sin u) 5 cos2 u.
Solution Use the distributive property to simplify the left side.
?
sin u (csc u 2 sin u) 5 cos2 u
?
sin u csc u 2 sin2 u 5 cos2 u
sin u A sin1 u B 2 sin2 u 5 cos2 u
?
1 2 sin2 u 5 cos2 u
?
?
(cos2 u 1 sin2 u) 2 sin2 u 5 cos2 u
cos u 5 cos u ✔
2
2
Use the Pythagorean identity
cos2 u + sin2 u 5 1.
The Pythagorean identity cos2 u 1 sin2 u 5 1 can be rewritten as
cos u 5 1 2 sin2 u. The second to last line of the proof is often omitted and
the left side, 1 2 sin2 u, replaced by cos2 u.
2
EXAMPLE 3
2
u
Prove the identity 1 2 sin u 5 1 cos
1 sin u.
Solution For this identity, it appears that we need to multiply both sides of the equation
by (1 1 sin u) to clear the denominator. However, in proving an identity we
perform only operations that change the form but not the value of that side of
the equation.
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487
Here we will work with the right side because it is more complicated and mulsin u
tiply by 11 2
2 sin u, a fraction equal to 1.
2
u
1 2 sin u 5 1 cos
1 sin u
?
2u
?
1 2 sin u
1 2 sin u 5 1 cos
1 sin u 3 1 2 sin u
?
cos2 u (1 2 sin u)
1 2 sin2 u
?
cos2 u (1 2 sin u)
cos2 u
1 2 sin u 5
1 2 sin u 5
1 2 sin u 5 1 2 sin u ✔
EXAMPLE 4
2
u
Prove the identity csccot
u 1 1 1 1 5 csc u.
Solution In this identity we will work with the left side.
How to Proceed
(1) Write the given equation:
(2) Write 1 as a fraction with the
same denominator as the given
fraction:
?
cot2 u
csc u 1 1 1 1 5
csc u 1 1 ?
cot2 u
csc u 1 1 1 csc u 1 1 5
(3) Add the fractions:
(4) Use the identity cot2 u 1 1 5 csc2 u:
(5) Factor the numerator:
(6) Divide the numerator and
denominator by (csc u 1 1):
csc u
csc u
cot2 u 1 csc u 1 1 ?
5 csc u
csc u 1 1
csc2 u 1 csc u ?
csc u 1 1 5 csc u
csc u (csc u 1 1) ?
5 csc u
csc u 1 1
csc u 5 csc u ✔
Exercises
Writing About Mathematics
1. Is sin u 5 "1 2 cos2 u an identity? Explain why or why not.
2
u
2. Cory said that in Example 3, 1 2 sin u 5 1 cos
1 sin u could have been shown to be an identity
sin u
by multiplying the left side by 11 1
1 sin u. Do you agree with Cory? Explain why or why not.
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Developing Skills
In 3–26, prove that each equation is an identity.
3. sin u csc u cos u 5 cos u
4. tan u sin u cos u 5 sin2 u
5. cot u sin u cos u 5 cos2 u
6. sec u (cos u 2 cot u) 5 1 2 csc u
7. csc u (sin u 1 tan u) 5 1 1 sec u
u
2
8. 1 2 cos
sec u 5 sin u
sin u
2
9. 1 2 csc
u 5 cos u
10. sin u (csc u 2 sin u) 5 cos2 u
11. cos u (sec u 2 cos u) 5 sin2 u
u
12. tan
sec u 5 sin u
u
13. cot
csc u 5 cos u
csc u
14. sec
u 5 cot u
u
15. sec
csc u 5 tan u
u
16. sin u1cos u 2 cos
sin u 5 tan u
sin2 u
18. 1 1
cos u 5 1 2 cos u
sin u
17. sin u1cos u 2 cos
u 5 cot u
cos2 u
19. 1 1 sin u 5 1 2 sin u
20. sec u csc u 5 tan u 1 cot u
tan2
sin2 u
22. cos u 1 1 1
cos u 5 1
21. sec u 2u 1 2 1 5 sec u
2u
23. sin u 1 1 cos
1 sin u 5 1
sec u
2
24. cos
u 2 tan u 5 1
u
sin u
26. cos
sec u 1 csc u 5 1
u
2
25. csc
sin u 2 cot u 5 1
sin u
u
27. For what values of u is the identity cos
sec u 1 csc u 5 1 undefined?
12-3 COSINE (A 2 B)
We can prove that cos (A 2 B) 5 cos A 2 cos B is not an identity by finding
one pair of values of A and B for which each side of the equation is defined and
the equation is false. For example, if, in degree measure, A 5 90° and B 5 60°,
?
cos (90° 2 60°) 5 cos 90° 2 cos 60°
cos 30° 5 0 2 21
?
!3
2
y
P(cos A, sin A)
Q(cos B, sin B)
A
B
O
x
2 212 ✘
In order to write an identity that expresses
cos (A 2 B) in terms of function values of A and B, we
will use the relationship between the unit circle and
the sine and cosine of an angle.
Let A and B be any two angles in standard position. The terminal side of A intersects the unit circle
at P(cos A, sin A) and the terminal side of B intersects the unit circle at Q(cos B, sin B). Use the distance formula to express PQ in terms of sin A, cos A,
sin B, and cos B:
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489
PQ2 5 (cos A 2 cos B)2 1 (sin A 2 sin B)2
5 (cos2 A 2 2 cos A cos B 1 cos2 B) 1 (sin2 A 2 2 sin A sin B 1 sin2 B)
5 (cos2 A 1 sin2 A) 1 (cos2 B 1 sin2 B) 2 2 cos A cos B 2 2 sin A sin B
5 1 1 1 2 2(cos A cos B 1 sin A sin B)
5 2 2 2(cos A cos B 1 sin A sin B)
y
P(cos (A 2 B), sin (A 2 B))
A
O
2
B
Q(1, 0)
x
Now rotate OQP through an angle of 2B, that is, an
angle of B units in the clockwise direction so that the image of
P is P9 and the image of Q is Q9. Q9 is a point on the x-axis
whose coordinates are (1, 0). Angle Q9OP9 is an angle in standard position whose measure is (A 2 B). Therefore, the coordinates of P9 are (cos (A 2 B), sin(A 2 B)). Use the distance
formula to find P9Q9.
(P9Q9)2 5 (cos (A 2 B) 2 1)2 1 (sin(A 2 B) 2 0)2
5 cos2 (A 2 B) 2 2 cos (A 2 B) 1 1 1 sin2(A 2 B)
5 [cos2 (A 2 B) 1 sin2(A 2 B)] 2 2 cos (A 2 B) 1 1
5 1 2 2 cos (A 2 B) 1 1
5 2 2 2 cos (A 2 B)
Distance is preserved under a rotation. Therefore,
(P9Q9)2 5 (PQ)2
2 2 2 cos (A 2 B) 5 2 2 2(cos A cos B 1 sin A sin B)
22 cos (A 2 B) 5 22(cos A cos B 1 sin A sin B)
cos (A 2 B) 5 cos A cos B 1 sin A sin B
At the beginning of this section we showed that:
cos (90° 2 60°) cos 90° 2 cos 60°
Does the identity that we proved make it possible to find cos (90° 2 60°)? We
can check.
cos (A 2 B) 5 cos A cos B 1 sin A sin B
?
cos (90° 2 60°) 5 cos 90° cos 60° 1 sin 90° sin 60°
cos 30° 5 0 A 12 B 1 1Q !3
2 R
?
!3 ?
!3
2 501 2
!3
!3
2 5 2 ✔
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EXAMPLE 1
Use (60° 2 45°) 5 15° to find the exact value of cos 15°.
Solution
cos (A 2 B) 5 cos A cos B 1 sin A sin B
cos (60° 2 45°) 5 cos 60° cos 45° 1 sin 60° sin 45°
!3 !2
cos 15° 5 A 12 B Q !2
2 R 1 Q 2 RQ 2 R
!6
cos 15° 5 !2
4 1 4
!6
Answer
cos 15° 5 !2 1
4
Note: Example 1 shows that cos 15° is an irrational number.
Cosine of (90° 2 B)
The cofunction relationship between cosine and sine can be proved using
cos (A 2 B). Use the identity for cos (A 2 B) to express cos (90° 2 B) in terms
of a function of B.
Let A 5 90°.
cos (A 2 B) 5 cos A cos B 1 sin A 1 sin B
cos (90° 2 B) 5 cos 90° cos B 1 sin 90° sin B
cos (90° 2 B) 5 0 cos B 1 1 sin B
cos (90° 2 B) 5 sin B
This is an identity, a statement true for all values of B.
EXAMPLE 2
Use the identity cos (90° 2 B) 5 sin B to find sin (90° 2 B).
Solution Let B 5 (90° 2 A).
cos (90° 2 B) 5 sin B
cos (90° 2 (90° 2 A)) 5 sin (90° 2 A)
cos (90° 2 90° 1 A) 5 sin (90° 2 A)
cos A 5 sin (90° 2 A) Answer
EXAMPLE 3
Given that A and B are second-quadrant angles, sin A 5 31, and sin B 5 15, find
cos (A 2 B).
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491
Solution Use the identity cos2 u 5 1 2 sin2 u to find cos A and cos B.
cos2 A 5 1 2 sin2 A
cos2 B 5 1 2 sin2 B
2
cos2 A 5 1 2 A 13 B
2
cos2 B 5 1 2 A 15 B
cos2 A 5 98
cos2 B 5 24
25
cos2 A 5 1 2 91
1
cos2 B 5 1 2 25
cos A 5 6#89
cos B 5 6#24
25
A is in the second quadrant.
B is in the second quadrant.
cos A 5 22 !2
3
cos B 5 22 !6
5
Use the identity for the cosine of the difference of two angles.
cos (A 2 B) 5 cos A cos B 1 sin A sin B
2 !6
1 1
cos (A 2 B) 5 Q22 !2
3 RQ2 5 R 1 A 3 B A 5 B
1
cos (A 2 B) 5 4 !12
15 1 15
1 1
cos (A 2 B) 5 4 !12
15
cos (A 2 B) 5 8 !3151 1
Note that since cos (A 2 B) is positive, (A 2 B) must be a first- or fourthquadrant angle.
Answer cos (A 2 B) 5 8 !3151
1
SUMMARY
We have proved the following identities:
cos (A 2 B) 5 cos A cos B 1 sin A sin B
cos (90° 2 B) 5 sin B
sin (90° 2 A) 5 cos A
Exercises
Writing About Mathematics
1. Are the equations sin u 5 cos (90° 2 u) and cos u 5 sin (90° 2 u) true for all real numbers
or only for values of u in the interval 0 , u , 90°?
2. Emily said that, without finding the values on a calculator, she knows that
sin 100° 5 cos (210°). Do you agree with Emily? Explain why or why not.
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Developing Skills
In 3–17, find the exact value of cos (A 2 B) for each given pair of values.
3. A 5 180°, B 5 60°
4. A 5 180°, B 5 45°
5. A 5 180°, B 5 30°
6. A 5 270°, B 5 60°
7. A 5 270°, B 5 30°
8. A 5 60°, B 5 90°
9. A 5 30°, B 5 90°
10. A 5 90°, B 5 60°
11. A 5 60°, B 5 270°
12. A 5 45°, B 5 270°
13. A 5 30°, B 5 270°
14. A 5 360°, B 5 60°
15. A 5 p, B 5 2p
3
16. A 5 p6 , B 5 4p
3
7p
17. A 5 3p
4 ,B5 4
Applying Skills
In 18–20, show all work.
18. a. Find the exact value of cos 15° by using cos (45° 2 30°).
b. Use the value of cos 15° found in a to find cos 165° by using cos (180° 2 15°).
c. Use the value of cos 15° found in a to find cos 345° by using cos (360° 2 15°).
d. Use cos A 5 sin (90° 2 A) to find the exact value of sin 75°.
19. a. Find the exact value of cos 120° by using cos (180° 2 60°).
b. Find the exact value of sin 120° by using cos2 u 1 sin2 u 5 1 and the value of cos 120°
found in a.
c. Find the exact value of cos 75° by using cos (120° 2 45°).
d. Use the value of cos 75° found in c to find cos 105° by using cos (180° 2 75°).
e. Use the value of cos 75° found in c to find cos 285° by using cos (360° 2 75°).
f. Find the exact value of sin 15°.
20. a. Find the exact value of cos 210° by using cos (270° 2 60°).
b. Find the exact value of sin 210° by using cos2 u 1 sin2 u 5 1 and the value of cos 210°
found in a.
c. Find the exact value of cos 165° by using cos (210° 2 45°).
d. Use the value of cos 165° found in c to find cos (215°) by using cos (165° 2 180°).
e. Use the value of cos (215°) found in d to find cos 195° by using cos (180° 2 (215°)).
f. Use the value of cos (215°) found in d to find the exact value of sin 105°.
21. A telephone pole is braced by two wires that are both fastened to the
ground at a point 15 feet from the base of the pole. The shorter wire is
fastened to the pole 15 feet above the ground and the longer wire 20
feet above the ground.
20 feet
15 feet
a. What is the measure, in degrees, of the angle that the shorter wire
makes with the ground?
15 feet
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493
b. Let u be the measure of the angle that the longer wire makes with the ground. Find
sin u and cos u.
c. Find the cosine of the angle between the wires where they meet at the ground.
d. Find, to the nearest degree, the measure of the angle between the wires.
12-4 COSINE (A + B)
We know that cos (A 1 B) can be written as cos (A 2 (2B)). Therefore,
cos (A 1 B) 5 cos (A 2 (2B)) 5 cos A cos (2B) 1 sin A sin (2B)
We would like to write this identity in term of cos B and sin B. Therefore,
we want to find the relationship between cos B and cos (2B) and between
sin A and sin (2A). In the exercises of Sections 11-1 and 11-2, we showed that
y 5 sin x is an odd function and y 5 cos x is an even function. That is, for all
values of x, 2sin x 5 sin (2x) and cos x 5 cos (2x). We can establish these
results graphically on the unit circle as the following hands-on activity
demonstrates.
Hands-On Activity
1. Draw a first-quadrant angle in standard
position. Let the point of intersection of
the initial side with the unit circle be Q
and the intersection of the terminal side
with the unit circle be P. Let the measure
of the angle be u; mQOP 5 u.
2. Reflect QOP in the x-axis. The
image of QOP is QOP9 and
mQOP9 5 2u.
3. Express sin u, cos u, sin (2u), and
cos (2u) as the lengths of line segments.
Show that cos (2u) 5 cos u and that
sin (2u) 5 2sin u.
y
P
u
O
u9
Q
x
P
4. Repeat steps 1 through 3 for a second-quadrant angle.
5. Repeat steps 1 through 3 for a third-quadrant angle.
6. Repeat steps 1 through 3 for a fourth-quadrant angle.
An algebraic proof can be used to prove the relationships cos (2u) 5 cos u
and sin (2u) 5 2sin u.
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Proof of cos (2u) 5 cos u
In the identity for cos (A 2 B), let A 5 0 and B 5 u.
cos (A 2 B) 5 cos A cos B 1 sin A sin B
cos (0 2 u) 5 cos 0 cos u 1 sin 0 sin u
cos (2u) 5 1 cos u 1 0 sin u
cos (2u) 5 cos u 1 0
cos (2u) 5 cos u ✔
Proof of sin (2u) 5 2sin u
In the identity sin A 5 cos (90° 2 A), let A 5 –u.
sin (2u) 5 cos (90° 2 (2u))
sin (2u) 5 cos (90° 1 u)
sin (2u) 5 cos (u 1 90°)
sin (2u) 5 cos (u 2 (290°))
sin (2u) 5 cos u cos (290°) 1 sin u sin (290°)
sin (2u) 5 (cos u)(0) 1 (sin u)(21)
sin (2u) 5 0 2 sin u
sin (2u) 5 2sin u ✔
Proof of the identity for cos (A 1 B)
cos (A 1 B) 5 cos (A 2 (2B))
cos (A 1 B) 5 cos A cos (2B) 1 sin A sin (2B)
cos (A 1 B) 5 cos A cos B 1 sin A (2sin B)
cos (A 1 B) 5 cos A cos B 2 sin A sin B
EXAMPLE 1
Find the exact value of cos 105° 5 cos (45° 1 60°) using identities.
Solution
cos (A 1 B) 5 cos A cos B 2 sin A sin B
cos (45° 1 60°) 5 cos 45° cos 60° 2 sin 45° sin 60°
!2 !3
1
cos 105° 5 Q !2
2 R A 2 B 2 Q 2 RQ 2 R
!6
cos 105° 5 !2
4 2 4
!6
Answer
cos 105° 5 !2 2
4
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495
EXAMPLE 2
Show that cos (p 1 u) 5 2cos u.
Solution We are now working in radians.
cos (A 1 B) 5 cos A cos B 2 sin A sin B
cos (p 1 u) 5 cos p cos u 2 sin p sin u
cos (p 1 u) 5 21 cos u 2 0 sin u
cos (p 1 u) 5 2cos u ✔
SUMMARY
We have proved the following identities:
cos (2u) 5 cos u
sin (2u) 5 2sin u
cos (A 1 B) 5 cos A cos B 2 sin A sin B
Exercises
Writing About Mathematics
1. Maggie said that cos (A 1 B) 1 cos (A 2 B) 5 cos 2A. Do you agree with Maggie? Justify
your answer.
2. Germaine said cos (A 1 B) 1 cos (A 2 B) 5 2 cos A cos B. Do you agree with Germaine?
Justify your answer.
Developing Skills
In 3–17, find the exact value of cos (A 1 B) for each given pair of values.
3. A 5 90°, B 5 60°
4. A 5 90°, B 5 45°
5. A 5 90°, B 5 30°
6. A 5 180°, B 5 60°
7. A 5 180°, B 5 30°
8. A 5 180°, B 5 45°
9. A 5 270°, B 5 30°
10. A 5 270°, B 5 60°
11. A 5 270°, B 5 45°
12. A 5 60°, B 5 60°
13. A 5 60°, B 5 270°
14. A 5 45°, B 5 270°
15. A 5 p2 , B 5 2p
3
16. A 5 p3 , B 5 4p
3
17. A 5 p4 , B 5 p6
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Applying Skills
In 18–20, show all work.
18. a. Find the exact value of cos 75° by using cos (45° 1 30°).
b. Use the value of cos 75° found in a to find cos 255° by using cos (180° 1 75°).
c. Use cos A 5 sin (90° 2 A) to find the exact value of sin 15°.
19. a. Find the exact value of cos 120° by using cos (60° 1 60°).
b. Find the exact value of sin 120° by using cos2 u 1 sin2 u 5 1 and the value of cos 120°
found in a.
c. Find the exact value of cos 165° by using cos (120° 1 45°).
d. Use the value of cos 165° found in c to find cos 345° by using cos (180° 1 165°).
20. a. Find the exact value of cos 315° by using cos (270° 1 45°).
b. Find the exact value of sin 315° by using cos2 u 1 sin2 u 5 1 and the value of cos 315°
found in a.
c. Find the exact value of cos 345° by using cos (315° 1 30°).
d. Explain why cos 405° 5 cos 45°.
21. An engineer wants to determine CD, the exact height of a building.
To do this, he first locates B on CD, a point 30 feet above C at the
foot of the building. Then he locates A, a point on the ground 40 feet
from C. From A, the engineer then finds that the angle of elevation
of D is 45° larger than u, the angle of elevation of B.
D
a. Find AB, sin u, and cos u.
b. Use sin u and cos u found in a to find the exact value of
cos (u 1 45°).
B
C A
c. Use the value of cos (u 1 45°) found in b to find AD.
d. Find CD, the height of the building.
12-5 SINE (A 2 B) AND SINE (A 1 B)
We can use the cofunction identity sin u 5 cos (90° 2 u) and the identities
for cos (A 2 B) and cos (A 1 B) to derive identities for sin (A 2 B) and
sin (A 1 B).
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Sine (A 2 B) and Sine (A 1 B)
Sine of (A 2 B)
Let u 5 A 2 B.
sin u 5 cos (90° 2 u)
sin (A 2 B) 5 cos (90° 2 (A 2 B))
5 cos (90° 2 A 1 B)
5 cos ((90° 2 A) 1 B)
5 cos (90° 2 A) cos B 2 sin (90° 2 A) sin B
5 sin A cos B 2 cos A sin B
sin (A 2 B) 5 sin A cos B 2 cos A sin B
Sine of (A 1 B)
The derivation of an identity for sin (A 1 B) is similar.
Let u 5 A 1 B.
sin u 5 cos (90° 2 u)
sin (A 1 B) 5 cos (90° 2 (A 1 B))
5 cos (90° 2 A 2 B)
5 cos ((90° 2 A) 2 B)
5 cos (90° 2 A) cos B 1 sin (90° 2 A) sin B
5 sin A cos B 1 cos A sin B
sin (A 1 B) 5 sin A cos B 1 cos A sin B
EXAMPLE 1
Use sin (45° 2 30°) to find the exact value of sin 15°.
Solution
sin (A 2 B) 5 sin A cos B 2 cos A sin B
sin (45° 2 30°) 5 sin 45° cos 30° 2 cos 45° sin 30°
!3
!2 1
sin 15° 5 Q !2
2 RQ 2 R 2 Q 2 RQ 2 R
!2
5 !6
4 2 4
!2
Answer
5 !6 2
4
497
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EXAMPLE 2
Use sin (60° 1 45°) to find the exact value of sin 105°.
sin (A 1 B) 5 sin A cos B 1 cos A sin B
Solution
sin (60° 1 45°) 5 sin 60° cos 45° 1 cos 60° sin 45°
!2
1 !2
sin 105° 5 Q !3
2 RQ 2 R 1 A 2 B Q 2 R
!2
5 !6
4 1 4
!2
5 !6 1
Answer
4
EXAMPLE 3
Show that sin (p 1 u) 5 2sin u.
Solution We are now working in radians.
sin (A 1 B) 5 sin A cos B 1 cos A sin B
sin (p 1 u) 5 sin p cos u 1 cos p sin u
sin (p 1 u) 5 0 cos u 1 (21) sin u
sin (p 1 u) 5 0 2 sin u
sin (p 1 u) 5 2sin u ✔
SUMMARY
We have proved the following identities:
sin (A 2 B) 5 sin A cos B 2 cos A sin B
sin (A 1 B) 5 sin A cos B 1 cos A sin B
Exercises
Writing About Mathematics
1. William said that sin (A 1 B) 1 sin (A 2 B) 5 sin 2A. Do you agree with William? Justify
your answer.
2. Freddy said that sin (A 1 B) 1 sin (A 2 B) 5 2 sin A cos B. Do you agree with Freddy?
Justify your answer.
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Sine (A 2 B) and Sine (A 1 B)
Developing Skills
In 3–17, find the exact value of sin (A 2 B) and of sin (A 1 B) for each given pair of values.
3. A 5 180°, B 5 60°
4. A 5 180°, B 5 45°
5. A 5 180°, B 5 30°
6. A 5 270°, B 5 60°
7. A 5 270°, B 5 30°
8. A 5 60°, B 5 90°
9. A 5 30°, B 5 90°
10. A 5 90°, B 5 60°
11. A 5 60°, B 5 270°
12. A 5 45°, B 5 270°
13. A 5 30°, B 5 270°
14. A 5 360°, B 5 60°
15. A 5 3p
2 , B 5 2p
p
16. A 5 2p
3 ,B5 6
17. A 5 p3 , B 5 5p
4
Applying Skills
In 18–20, show all work.
18. a. Find the exact value of sin 15° by using sin (45° 2 30°).
b. Use the value of sin 15° found in a to find sin 165° by using sin (180° 2 15°).
c. Use the value of sin 15° found in a to find sin 345° by using sin (360° 2 15°).
d. Use the value of sin 15° found in a to find sin 195° by using sin (180° 1 15°).
19. a. Find the exact value of sin 120° by using sin (180° 2 60°).
b. Find the exact value of cos 120° by using cos (180° 2 60°).
c. Find the exact value of sin 75° by using sin (120° 2 45°).
d. Use the value of sin 75° found in c to find sin 105° by using sin (180° 2 75°).
e. Use the value of sin 75° found in c to find sin 285° by using sin (360° 2 75°).
20. a. Find the exact value of sin 210° by using sin (270° 2 60°).
b. Find the exact value of cos 210° by using cos (270° 2 60°).
c. Find the exact value of sin 165° by using sin (210° 2 45°).
d. Use the value of sin 165° found in c to find sin (215°) by using sin (165° 2 180°).
e. Use the value of sin (215°) found in d to find sin 195° by using sin (180° 2 (215°)).
f. Use the value of sin (215°) found in d to find the exact value of sin 105°.
21. A telephone pole is braced by two wires that are both fastened to the ground at the same point. The shorter wire is
30 feet long and is fastened to the pole 10 feet above the
foot of the pole. The longer wire is 33 feet long and is fastened to the pole 17 feet above the foot of the pole.
a. If the measure of the angle that the longer wire makes
with the ground is x and the measure of the angle that
the shorter wire makes with the ground is y, find sin x,
cos x, sin y, and cos y.
17 feet
t
fee
33 feet
30
b. Find the exact value of sin (x 2 y), the sine of the angle between the two wires.
c. Find to the nearest degree the measure of the angle between the two wires.
10 feet
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22. Two boats leave the same dock to cross a river that is 500 feet wide. The first boat leaves the
dock at an angle of u with the shore and travels 1,300 feet to reach a point downstream on
the opposite shore of the river. The second boat leaves the dock at an angle of (u 1 30°)
with the shore.
a. Find sin u and cos u.
b. Find sin (u 1 30°).
c. Find the distance that the second boat travels to reach the opposite shore.
500 ft
t
0f
0
1,3
2
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23. The coordinates of any point in the coordinate plane can be written as A(r cos a, r sin a)
h
when a is the measure of the angle between the positive ray of the x-axis and OA and
r 5 OA. Under a rotation of u about the origin, the coordinates of A9, the image of A, are
(r cos (a 1 u), r sin (a 1 u)). Find the coordinates of A9, the image of A(6, 8), under a rotation of 45° about the origin.
12-6 TANGENT (A – B) AND TANGENT (A + B)
sin u
We can use the identity tan u 5 cos
u and the identities for sin (A 1 B) and
cos (A 1 B) to write identities for tan (A 1 B) and tan (A 2 B).
Tangent of (A 1 B)
sin (A 1 B)
sin A cos B 1 cos A sin B
tan (A 1 B) 5 cos (A 1 B) 5 cos
A cos B 2 sin A sin B
We would like to write this identity for tan (A 1 B) in terms of tan A and
tan B. We can do this by dividing each term of the numerator and each term of
the denominator by cos A cos B. When we do this we are dividing by a fraction
equal to 1 and therefore leaving the value of the expression unchanged.
sin A cos B
cos A sin B
sin A cos B
cos A sin B
cos A cos B
cos A cos B
cos A cos B
cos A cos B
A cos B 1 cos A cos B
A cos B 1 cos A cos B
tan (A 1 B) 5 cos
5 cos
cos A cos B 2 sin A sin B
cos A cos B 2 sin A sin B
A 1 tan B
tan (A 1 B) 5 1tan
2 tan A tan B
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501
Tangent of (A 2 B)
The identity for tan (A – B) can be derived in a similar manner.
sin (A 2 B)
sin A cos B 2 cos A sin B
tan (A 2 B) 5 cos (A 2 B) 5 cos
A cos B 1 sin A sin B
sin A cos B
cos A sin B
sin A cos B
cos A sin B
cos A cos B
cos A cos B
cos A cos B
cos A cos B
A cos B 2 cos A cos B
A cos B 2 cos A cos B
tan (A 2 B) 5 cos
5 cos
cos A cos B 1 sin A sin B
cos A cos B 1 sin A sin B
A 2 tan B
tan (A 2 B) 5 1tan
1 tan A tan B
These identities are true for all replacements of A and B for which
cos A 0 and cos B 0, and for which tan (A B) or tan (A B) are defined.
EXAMPLE 1
Use tan 2p 5 0 and tan p4 5 1 to show that tan 7p
4 5 21.
A 2 tan B
tan (A 2 B) 5 1tan
1 tan A tan B
Solution
4
tan A 2p 2 p4 B 5
1 1 tan 2p tan p
4
tan 2p 2 tan p
0 2 1
tan 7p
4 5 1 1 (0)(1)
tan 7p
4 5 21 ✔
EXAMPLE 2
Use (45° 1 120°) 5 165° to find the exact value of tan 165°.
!2
!2
2 and cos 45° 5 2 , so tan 45° 5 1.
1
5 !3
2 and cos 120° 5 22 , so tan 120° 5
Solution sin 45° 5
sin 120°
2!3.
A 1 tan B
tan (A 1 B) 5 1tan
2 tan A tan B
458 1 tan 1208
tan (458 1 1208) 5 1tan
2 tan 458 tan 1208
1 1 A2!3B
tan 1658 5 1 2 (1) A2!3B
!3
!3
5 11 2
3 11 2
1 !3
2 !3
tan 1658 5 4 2222 !3 5 22 1 !3
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Alternative Write the identity tan (A 1 B) in terms of sine and cosine.
Solution
A cos B 1 cos A sin B
tan (A 1 B) 5 sin
cos A cos B 2 sin A sin B
458 cos 1208 1 cos 458 sin 1208
tan (458 1 1208) 5 sin
cos 458 cos 1208 2 sin 458 sin 1208
tan 1658 5
5
!2 !3
1
Q !2
2 RQ22 R 1 Q 2 RQ 2 R
!2 !3
1
Q !2
2 RQ22 R 2 Q 2 RQ 2 R
4
!6
2 !2
4 1 4 3 !2
4
!6
2 !2
!2
4 2 4
1 !3
tan 1658 5 21
5 22 1 !3
21 2 !3
Answer tan 165° 5 22 1 !3 or !3 2 2
SUMMARY
We have proved the following identities:
A 1 tan B
tan (A 1 B) 5 1tan
2 tan A tan B
A 2 tan B
tan (A 2 B) 5 1tan
1 tan A tan B
Exercises
Writing About Mathematics
A 1 tan B
1. Explain why the identity tan (A 1 B) 5 1tan
2 tan A tan B is not valid when A or B is equal to
p
2 1 np for any integer n.
p
A 1 tan B
p
2. Explain why 1tan
2 tan A tan B is undefined when A 5 6 and B 5 3 .
Developing Skills
In 3–17, find the exact value of tan (A 1 B) and of tan (A 2 B) for each given pair of values.
3. A 5 45°, B 5 30°
4. A 5 45°, B 5 60°
5. A 5 60°, B 5 60°
6. A 5 180°, B 5 30°
7. A 5 180°, B 5 45°
8. A 5 180°, B 5 60°
9. A 5 120°, B 5 30°
10. A 5 120°, B 5 45°
11. A 5 120°, B 5 60°
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Tangent (A 2 B) and Tangent (A 1 B)
12. A 5 120°, B 5 120°
15. A 5 p, B 5
13. A 5 240°, B 5 120°
p
3
16. A 5
5p
6 ,B
5
14. A 5 360°, B 5 60°
5p
6
17. A 5 p3 , B 5 p4
Applying Skills
18. Prove that tan (180° 1 u) 5 tan u.
19. Find tan (A 1 B) if tan A 5 3 and tan B 5 212.
20. Find tan (A 2 B) if tan A 5 34 and tan B 5 28.
21. Find tan (A 1 B) if A is in the second quadrant, sin A 5 0.6, and tan B 5 4.
22. If A 5 arctan 2 and B 5 arctan (22), find tan (A 2 B).
23. If A 5 arctan A 223 B and B 5 arctan 23, find tan (A 1 B).
24. In the diagram, ABCD, BEFC, and EGHF are congruent squares with AD 5 1. Let
mGAH 5 x, mGBH 5 y, and mGEH 5 z.
D
C
H
F
1
y
x
A
B
z
E
503
G
a. Find tan (x 1 y).
b. Does x 1 y 5 z? Justify your answer.
25. A tower that is 20 feet tall stands at the edge of a 30-foot
cliff. From a point on level ground that is 20 feet from a
point directly below the tower at the base of the cliff, the
measure of the angle of elevation of the top of the tower
is x and the measure of the angle of elevation of the foot
of the tower is y.
20 ft
a. Find the exact value of tan (x 2 y), the tangent of the
angle between the lines of sight to the foot and top of
the tower.
b. Find to the nearest degree the measure of the angle
between the lines of sight to the foot and the top of the
tower.
30 ft
20 ft
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Trigonometric Identities
26. Two boats leave a dock to cross a river that is 80 meters wide. The first boat travels to a
point that is 100 meters downstream from a point directly opposite the starting point, and
the second boat travels to a point that is 200 meters downstream from a point directly opposite the starting point.
a. Let x be the measure of the angle between the river’s edge and the path of the first
boat and y be the measure of the angle between the river’s edge and the path of the
second boat. Find tan x and tan y.
b. Find the tangent of the measure of the angle between the paths of the boats.
80 m
2
1
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200 m
100 m
12-7 FUNCTIONS OF 2A
The identities for sin (A 1 B), cos (A 1 B), and tan (A 1 B) are true when
A = B. These identities can be used to find the function values of 2A. We often
call the identities used to find the function values of twice an angle double-angle
formulas.
Sine of 2A
In the identity for sin (A 1 B), let B 5 A.
sin (A 1 B) 5 sin A cos B 1 cos A sin B
sin (A 1 A) 5 sin A cos A 1 cos A sin A
sin (2A) 5 2 sin A cos A
Cosine of 2A
In the identity for cos (A 1 B), let B 5 A.
cos (A 1 B) 5 cos A cos B 2 sin A sin B
cos (A 1 A) 5 cos A cos A 2 sin A sin A
cos (2A) 5 cos2 A 2 sin2 A
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Functions of 2A
505
The identity for cos 2A can be written in two other ways:
cos (2A) 5 2 cos2 A 2 1
cos (2A) 5 1 2 2 sin2 A
Proofs of the first of these identities are given in Example 2. The proof of
the second identity is left to the student in Exercise 23.
Tangent of 2A
In the identity for tan (A 1 B), let B 5 A.
A 1 tan B
tan (A 1 B) 5 1tan
2 tan A tan B
A 1 tan A
tan (A 1 A) 5 1tan
2 tan A tan A
2 tan A
tan (2A) 5 1 2
tan2 A
EXAMPLE 1
If tan u 5 2 !7
3 and u is a second-quadrant angle, find:
a. sec u
b. cos u
c. sin u
d. sin 2u
e. cos 2u
f. tan 2u
g. In what quadrant does 2u lie?
Solution a. Use the identity 1 1 tan2 u 5 sec2 u
to find sec u. Since u is a secondquadrant angle, sec u is negative.
2
1 1 Q2 !7
3 R 5 sec u
2
b. Use sec u 5 cos1 u to find cos u.
243 5 cos1 u
234 5 cos u Answer
1 1 79 5 sec2 u
16
9
243
2
5 sec2 u
5 sec u Answer
2
c. Use cos u 1 sin u 5 1 to find sin u.
Since u is a second-quadrant
angle, sin u will be positive.
A 234 B
2
9
16
1 sin2 u 5 1
1 sin2 u 5 1
7
sin2 u 5 16
sin u 5 !7
4 Answer
d. sin 2u 5 2 sin u cos u
3
5 2Q !7
4 RQ24 R
5 23 !7
8 Answer
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Trigonometric Identities
e. cos 2u 5 cos2 u 2 sin2 u
2
2
5 Q234 R 2 Q !7
4 R
2 tan u
f. tan 2u 5 1 2
tan2 u
5
9
7
2 16
5 16
2
5 18 Answer
5 16
5
2Q2 !7
3 R
2
1 2 Q2 !7
3 R
22 !7
3
5 26!7
2 5 23"7
2
9
Answer
g. Since sin 2u and tan 2u are negative and cos 2u is positive, 2u must be a
fourth-quadrant angle. Answer
EXAMPLE 2
Prove the identity cos 2u 5 2 cos2 u 2 1.
Solution First write the identity for cos 2u that we have already proved. Then substitute
for sin2 u using the Pythagorean identity sin2 u 5 1 2 cos2 u.
cos 2u 5 cos2 u 2 sin2 u
5 cos2 u 2 (1 2 cos2 u)
5 cos2 u 2 1 1 cos2 u
5 2 cos2 u 2 1 ✔
Alternative To the right side of the identity for cos 2u, add 0 in the form cos2 u 2 cos2 u.
Solution
cos 2u 5 cos2 u 2 sin2 u
5 cos2 u 2 sin2 u 1 cos2 u 2 cos2 u
5 cos2 u 1 cos2 u 2 (sin2 u 1 cos2 u)
5 2 cos2 u 2 1 ✔
SUMMARY
We have proved the following identities:
sin 2A 5 2 sin A cos A
cos 2A 5 cos2 A 2 sin2 A
cos 2A 5 2 cos2 A 2 1
cos 2A 5 1 2 2 sin2 A
2 tan A
tan 2A 5 1 2
tan2 A
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Functions of 2A
507
Exercises
Writing About Mathematics
1. Does cos 2u 5 sin 2(90° 2 u)? Justify your answer.
sin 2u
2. Does tan 2u 5 cos
2u? Justify your answer.
Developing Skills
In 3–8, for each value of u, use double-angle formulas to find a. sin 2u, b. cos 2u, c. tan 2u. Show all
work.
3. u 5 30°
6. u 5
p
4
4. u 5 225°
7. u 5
5. u 5 330°
8. u 5 5p
3
7p
6
In 9–20, for each given function value, find a. sin 2u, b. cos 2u, c. tan 2u, d. the quadrant in which 2u
lies. Show all work.
9. tan u 5 35, u in the first quadrant
11. sin u 5 2 !6
7 in the first quadrant
13. cos u 5 2 !10
7 in the first quadrant
10. tan u 5 2 !11
5 in the second quadrant
12. sin u 5 20.5 in the third quadrant
14. cos u 5 22 !5
5 in the second quadrant
15. tan u 5 12
5 in the third quadrant
16. sin u 5 2 !2
3 in the third quadrant
19. cot u 5 213 in the second quadrant
20. tan u 5 2 in the third quadrant
17. csc u 5 !5 in the second quadrant
18. sec u 5 !13
2 in the fourth quadrant
Applying Skills
2u
21. Prove the identity: cot u 5 1 2sincos
2u.
2u
sin 2u
22. Prove the identity: cos
sin u 1 cos u 5 csc u.
23. Show that cos 2u 5 1 2 2 sin2 u.
24. Show that csc 2u 5 12 sec u csc u.
25. a. Derive an identity for sin 4A in terms of the functions of 2A.
b. Derive an identity for cos 4A in terms of the functions of 2A.
c. Derive an identity for tan 4A in terms of the functions of 2A.
Hint: Let 4A 5 2u.
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Trigonometric Identities
26. A park in the shape of a rectangle, ABCD, is
crossed by two paths: AC, a diagonal, and AE,
which intersects BC at E. The measure of BAC is
twice the measure of BAE, AB 5 10 miles and
AE 5 12 miles.
A
10 mi
a. Let mBAE 5 u. Find the exact value of cos u.
B
b. Express mBAC in terms of u and find the exact
value of the cosine of BAC.
D
12 mi
E
C
c. Find the exact value of AC.
27. The image of A(4, 0) under a rotation of u about the origin is A9 A !10, !6B . What are the
coordinates of A0, the image of A9 under the same rotation?
12-8 FUNCTIONS OF 1
2A
Just as there are identities to find the function values of 2A, there are identities
to find cos 12A, sin 12A, and tan 12A. We often call the identities used to find the
function values of half an angle half-angle formulas.
Cosine of 1
2A
We can use the identity for cos 2u to write an identity for 12A. Begin with the
identity for cos 2u written in terms of cos u. Then solve for cos u.
cos 2u 5 2cos2 u 2 1
1 1 cos 2u 5 2cos2 u
1 1 cos 2u
2
1 1 cos 2u
6#
2
5 cos2 u
5 cos u
cos u 5 6#1 1 2cos 2u
Let 2u 5 A and u 5 12A.
cos 12A 5 6#1 1 2cos A
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Functions of 12A
509
Sine of 1
2A
Begin with the identity for cos 2u, this time written in terms of sin u. Then solve
for sin u.
cos 2u 5 1 2 2 sin2 u
2 sin2 u 5 1 2 cos 2u
sin2 u 5 1 2 2cos 2u
Let 2u 5 A and u 5 12A.
sin u 5 6#1 2 2cos 2u
sin 12A 5 6#1 2 2cos A
Tangent of 1
2A
sin u
1
1
Use the identity tan u 5 cos
u. Let u 5 2A. Then substitute in the values of sin 2A
1
and cos 2A.
tan 12A 5
tan 12A 5
tan 12A 5
sin 21A
cos 21A
6#1 2 2cos A
6#1 1 2cos A
!2
6#1 2 2cos A
3 1
!2
6#1 1 2cos A
1
!1 2 cos A
tan 12A 5 6 !1
1 cos A
When we use the identities for the function values of (A 1 B), (A 2 B), and
2A, the sign of the function value is a result of the computation. When we use
the identities for the function values of 12A, the sign of the function value must
be chosen according to the quadrant in which 12A lies.
For example, if A is a third-quadrant angle such that 180° , A , 270°, then
90° , 12A , 135°. Therefore, 12A is a second-quadrant angle. The sine value of 12A
is positive and the cosine and tangent values of 12A are negative.
If A is a third-quadrant angle and 540° , A , 630°, then 270° , 12A , 315°.
Therefore, 12A is a fourth-quadrant angle. The cosine value of 12A is positive and
the sine and tangent values of 12A are negative.
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Trigonometric Identities
EXAMPLE 1
If 180° , A , 270° and sin A 5 2 !5
3 , find:
a. sin 12A
b. cos 12A
c. tan 12A
Solution The identities for the function values of 12A are given in terms of cos A. Use
cos A 5 6"1 2 sin2 A to find cos A. Since A is a third-quadrant angle, cos A
is negative.
2
9
5
4
2
cos A 5 2%1 2 Q !5
3 R 5 2#9 2 9 5 2#9 5 23
If 180° , A , 270°, then 90° , 12A , 135°. Therefore, 12A is a second-quadrant angle: sin 12A is positive and cos 12A and tan 12A are negative. Use the halfangle identities to find the function values of 12A.
a. sin 12A 5 #1 2 2cos A
5
1 2 A 223 B
Å
2
b. cos 12A 5 2#1 1 2cos A
5 #56
52
Å
1 1 A 223 B
2
5 2#16
5 2 !6
6
5 !30
6
cos A
c. tan 12A 5 2#11 2
1 cos A
52
ã
1 2 A 223 B
1 1 Q223 R
5 2#51
5 2!5
Answers a. sin 12A 5
!30
6
b. cos 12A 5 2 !6
6
EXAMPLE 2
A
Show that tan 12A 5 61 1sincos
A.
c. tan 12A 5 2!5
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Functions of 12A
511
cos A
tan 12A 5 6#11 2
1 cos A
Solution
cos A
1 1 cos A
5 6#11 2
1 cos A 3 1 1 cos A
2 cos A
5 6#(11 1
cos A) 2
2
2A
5 6#(1 1sincos
A) 2
A
5 61 1sincos
A✔
SUMMARY
We have proved the following identities:
cos 12A 5 6#1 1 2cos A
sin 12A 5 6#1 2 2cos A
2 cos A
tan 12A 5 6#11 1
cos A
Exercises
Writing About Mathematics
1. Karla said that if cos A is positive, then 2p2 , A , p2 , 2p4 , 12A , p4 , and cos 12A is positive.
Do you agree with Karla? Explain why or why not.
2. In Example 1, can tan 12A be found by using
sin 21A
?
cos 21A
Explain why or why not.
Developing Skills
In 3–8, for each value of u, use half-angle formulas to find a. sin 12u b. cos 12u c. tan 21u. Show all work.
3. u 5 480°
6. u 5 2p
4. u 5 120°
7. u 5
7p
2
5. u 5 300°
8. u 5 3p
2
In 9–14, for each value of cos A, find a. sin 12A b. cos 12A c. tan 12A. Show all work.
9. cos A 5 34, 0° , A , 90°
5
10. cos A 5 212
, 90° , A , 180°
11. cos A 5 219, 180° , A , 270°
12. cos A 5 81, 270° , A , 360°
13. cos A 5 215, 450° , A , 540°
14. cos A 5 79, 360° , A , 450°
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Trigonometric Identities
1
1
1
15. If sin A 5 24
25 and 90° , A , 180°, find: a. sin 2A b. cos 2A c. tan 2A
16. If sin A 5 245 and 180° , A , 270°, find: a. sin 12A b. cos 12A c. tan 21A
1
1
1
17. If sin A 5 224
25 and 540° , A , 630°, find: a. sin 2A b. cos 2A c. tan 2A
18. If tan A 5 3 and 180° , A , 270°, find: a. sin 12A b. cos 12A c. tan 12A
Applying Skills
A
19. Show that tan 21A 5 61 2sincos
A .
p
20. Use cos A 5 cos p4 5 !2
2 to show that the exact value of tan 8 5 !2 2 1.
21. Use cos A 5 cos 30° 5 !3
2 to show that the exact value of tan 15° 5 2 2 !3.
"2 2 !3
22. Use cos A 5 cos 30° 5 !3
.
2 to show that the exact value of sin 15° 5
2
23. a. Derive an identity for sin 14A in terms of cos 12A.
b. Derive an identity for cos 14A in terms of cos 12A.
c. Derive an identity for tan 14A in terms of cos 12A.
Hint: Let 14A 5 12u.
24. The top of a billboard that is mounted on a base is
60 feet above the ground. At a point 25 feet from the foot
of the base, the measure of the angle of elevation to the
top of the base is one-half the measure of the angle of
elevation to the top of the billboard.
60 ft
This Billboard
for Lease
a. Let u be the measure of the angle of elevation to the
top of the billboard. Find the cos u and tan 12u.
b. Find the height of the base and the height of the billboard.
25 ft
Hands-On Activity:
Graphical Support for the Trigonometric Identities
We can use the graphing calculator to provide support that an identity is true. Treat each side
of the identity as a function and graph each function on the same set of axes. If the graphs of
the functions coincide, then we have provided graphical support that the identity is true. Note
that support is not the same as proof. In order to prove an identity, we need to use the algebraic methods from this chapter or similar algebraic methods.
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Chapter Summary
For example, to provide support for the identity sin2 u 1 cos2 u 5 1,
treat each side as a function. In your graphing calculator, enter
Y1 5 sin2 u 1 cos2 u and Y2 5 1. Graph both functions in the interval
22p # u # 2p. As the graph shows, both functions appear to
coincide.
For 1–3, explore each equation on the graphing calculator. Do
these equations appear to be identities?
513
Y1 5 sin2 u 1 cos2 u
Y2 5 1
1. cot u sin 2u 5 1 1 cos 2u
sin 3u
2. sin3 u 5 3 sin u 2
4
3. tan 2 u cos 2 u 5 1 2 cos 2u
Y2
4. Kevin tried to provide graphical support for the identity
cos 12A 5 6#1 1 2cos A by graphing the functions
1 1 cos X
Y1 5 cos A X
, Y3 5 2#1 1 2cos X in the
2 B , Y2 5 #
2
Y1
interval 0 # X # 2p. The graphs did not coincide, as shown
on the right. Explain to Kevin what he did wrong.
Y3
CHAPTER SUMMARY
A trigonometric identity can be proved by using the basic identities to
change one side of the identity into the form of the other side.
The function values of (A 6 B), 2A, and 12A, can be written in terms of the
function values of A and of B, using the following identities:
Sums of Angle Measures
Differences of Angle Measures
cos (A 1 B) 5 cos A cos B 2 sin A sin B
cos (A – B) 5 cos A cos B 1 sin A sin B
sin (A 1 B) 5 sin A cos B 1 cos A sin B
sin (A – B) 5 sin A cos B 2 cos A sin B
A 1 tan B
tan (A + B) 5 1tan
2 tan A tan B
A 2 tan B
tan (A – B) 5 1tan
1 tan A tan B
Double-Angle Formulas
sin (2A) 5 2 sin A cos A
2
2
cos (2A) 5 cos A 2 sin A
cos (2A) 5 2 cos2 A 2 1
cos (2A) 5 1 2 2 sin2 A
A
tan (2A) 5 1 22tan
tan2 A
Half-Angle Formulas
sin 12A 5 6#1 2 2cos A
cos 12A 5 6#1 1 2cos A
2 cos A
tan 12A 5 6#11 1
cos A
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VOCABULARY
12-7 Double-angle formulas
12-8 Half-angle formulas
REVIEW EXERCISES
In 1–6, prove each identity.
1. sec u 5 csc u tan u
2. cos u cot u 1 sin u 5 csc u
3. 2 sin2 u 5 1 2 cos 2u
cos u
4. tan u 1 csc1 u 5 1 1
cot u
2u 1 sin u
5. cossin
2u 1 cos u 1 1 5 tan u
6. sin (A 1 B) 2 sin (A 2 B) 5 tan A cot B
sin (A 1 B) 1 sin (A 2 B)
7
In 7–21, sin A 5 225
, sin B 5 235, and both A and B are third-quadrant angles.
Find each function value.
7. cos A
8. cos B
9. tan A
10. tan B
11. sin (A 1 B)
12. sin (A 2 B)
13. cos (A 1 B)
14. cos (A 2 B)
15. tan (A 1 B)
16. tan (A 2 B)
17. sin 2A
18. cos 2B
21. cos 12A
22. If cos A 5 0.2 and A and B are complementary angles, find cos B.
19. tan 2A
20. sin
1
2A
23. If sin A 5 0.6 and A and B are supplementary angles, find cos B.
In 24–32, u is the measure of an acute angle and sin u 5 !7
4 . Find each function
value.
24. cos u
25. tan u
26. sin 2u
27. cos 2u
28. tan 2u
30. cos 12u
31. tan 12u
29. sin 12u
32. cos (2u 1 u)
In 33–41, 360° , u , 450° and tan u 5 17. Find, in simplest radical form, each
function value.
33. sec u
34. cos u
35. sin u
36. cot u
37. sin 2u
38. cos 2u
41. cos 12u
42. Show that if A and B are complementary, cos A cos B 5 sin A sin B.
39. tan 2u
40. sin
1
2u
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Cumulative Review
515
Exploration
In this activity, you will derive the triple-angle formulas.
sin (3A) 5 3 sin A 2 4 sin3 A
cos (3A) 5 4 cos3 A 2 3 cos A
3
A 2 tan A
tan (3A) 5 3 tan
1 2 3 tan 2 A
1. Examine the graphs of these equations on the graphing calculator in
the interval 0 # A # 2p. Explain why these equations appear to be
identities.
2. Use 2A 1 A 5 3A to prove that each equation is an identity.
CUMULATIVE REVIEW
CHAPTERS 1–12
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. The solution set of 2 2 !x 1 3 5 6 is
(1) {1}
(2) {219}
(3) {13}
(4) [
2. The radian measure of an angle of 240° is
(1) 4p
3
(2) 2p
3
(3) 5p
6
(4) 7p
6
(3) 3
(4) 4
2
3. If 3x 5 273, then x is equal to
(1) 1
(2) 2
4. When expressed in a 1 bi form, A 12 1 !29 B 2 A 3 2 !4 B is equal to
(1) 10 1 3i
(2) 9 1 5i
(3) 11 1 3i
(4) 9 1 i
3
5. a 2n is equal to
n50
(1) 8
(2) 9
!7
6. The fraction 33 1
is equal to
2 !7
(1) 8 1 3 !7
(2) 8
(3) 14
(3) 8 1 !7
(4) 8 1 12 !7
(4) 15
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Trigonometric Identities
7. If f(x) 5 (x 1 2)2 and g(x) 5 x 2 1, then g(f(x)) 5
(1) (x 1 2)2 2 1
(3) x2 1 5x 1 3
2
(2) (x 1 1)
(4) (2x 1 1)2
8. The solution set of 2x2 2 5x 5 3 is
(3) U32, 1 V
(1) U12, 23 V
(2) U212, 3 V
(4) U 232, 21 V
9. When log x 5 2 log A 2 12 log B, x is equal to
2
(1) 2A 2 12B
A
(3) !B
(2) A2 2 !B
(4) 2A
1
2B
10. If y 5 arccos 0, then y is equal to
(1) 0
(2) 1
(3) p2
(4) p
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Write the multiplicative inverse of 1 1 i in a 1 bi form.
12. For what values of c and a does x2 1 5x 1 c 5 (x 1 a)2? Justify your
answer.
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Solve x2 1 3x 2 10 $ 0 for x and graph the solution set on a number
line.
14. Write the equation of a circle if the endpoints of a diameter of the
circle are (0, –1) and (2, 5).
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Cumulative Review
517
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. a. Sketch the graph of f(x) 5 sin x in the interval 22p # x # 2p.
b. On the same set of axes, sketch the graph of g(x) 5 f A x 1 p4 B 1 3 and
write an equation for g(x).
16. The first term of a geometric sequence is 1 and the fifth term is 9.
a. What is the common ratio of the sequence?
b. Write the first eight terms of the sequence.
c. Write the sum of the first eight terms of the sequence using sigma
notation.
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CHAPTER
13
CHAPTER
TABLE OF CONTENTS
13-1 First-Degree Trigonometric
Equations
13-2 Using Factoring to Solve
Trigonometric Equations
13-3 Using the Quadratic Formula
to Solve Trigonometric
Equations
13-4 Using Substitution to Solve
Trigonometric Equations
Involving More Than One
Function
13-5 Using Substitution to Solve
Trigonometric Equations
Involving Different Angle
Measures
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
518
TRIGONOMETRIC
EQUATIONS
The triangle is a rigid figure, that is, its shape cannot
be changed without changing the lengths of its sides.
This fact makes the triangle a basic shape in construction. The theorems of geometry give us relationships
among the measures of the sides and angles of triangle.
Precisely calibrated instruments enable surveyors to
obtain needed measurements. The identities and formulas of trigonometry enable architects and builders to
formulate plans needed to construct the roads, bridges,
and buildings that are an essential part of modern life.
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13-1 FIRST-DEGREE TRIGONOMETRIC EQUATIONS
A trigonometric equation is an equation whose variable is expressed in terms of
a trigonometric function value. To solve a trigonometric equation, we use the
same procedures that we used to solve algebraic equations. For example, in the
equation 4 sin u 1 5 5 7, sin u is multiplied by 4 and then 5 is added. Thus, to
solve for sin u, first add the opposite of 5 and then divide by 4.
4 sin u 1 5 5 7
2 5 5 25
4 sin u 5
4 sin u
4
5
sin u 5
2
2
4
1
2
We know that sin 30° 5 12, so one value of u is 30°; u1 5 30. We also know
that since sin u is positive in the second quadrant, there is a secondquadrant angle, u2, whose sine is 21. Recall the relationship between an angle in
any quadrant to the acute angle called the reference angle. The following table
compares the degree measures of u from 290° to 360°, the radian measures of
u from 2p2 to 2p, and the measure of its reference angle.
Angle
Reference
Angle
Fourth
Quadrant
First
Quadrant
Second
Quadrant
Third
Quadrant
Fourth
Quadrant
290° , u , 0°
2p2 , u , 0
0° , u , 90°
0 , u , p2
90° , u , 180°
p
2 ,u,p
180° , u , 270°
p , u , 3p
2
270° , u , 360°
3p
2 , u , 2p
2u
2u
u
u
180° 2 u
p2u
u 2 180°
u2p
360° 2 u
2p 2 u
The reference angle for the second-quadrant angle whose sine is 21 has a
degree measure of 30° and u2 5 180° 2 30° or 150°. Therefore, sin u2 5 12. For
0° u , 360°, the solution set of 4 sin u 1 5 5 7 is {30°, 150°}. In radian measure, the solution set is U p6 , 5p
6 V.
In the example given above, it was possible to give the exact value of u that
makes the equation true. Often it is necessary to use a calculator to find an
approximate value. Consider the solution of the following equation.
5 cos u 1 7 5 3
5 cos u 5 24
cos u 5 245
u 5 arccos A 245 B
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When we use a calculator to find u, the calculator will return the value of
the function y 5 arccos x whose domain is 0° x 180° in degree measure or
0 x p in radian measure.
In degree measure:
ENTER:
2nd
5
)
COS1
(-)
4 DISPLAY:
ENTER
To the nearest degree, one value of u is
143°. In addition to this second-quadrant
angle, there is a third-quadrant angle such
that cos u 5 245. To find this third-quadrant
angle, find the reference angle for u.
Let R be the measure of the reference
angle of the second-quadrant angle. That
is, R is the acute angle such that
cos u 5 2cos R.
cos-1(-4/5)
143.1301024
y
37°
143°
x
R 5 180° 2 u 5 180° 2 143° 5 37°
The measure of the third-quadrant angle is:
u 5 R 1 180°
u 5 37° 1 180°
u 5 217°
For 0° u 360°, the solution set of 5 cos u 1 7 5 3 is {143°, 217°}.
If the value of u can be any angle measure, then for all integral values of n,
u 5 143 1 360n or u 5 217 1 360n.
Procedure
To solve a linear trigonometric equation:
1. Solve the equation for the function value of the variable.
2. Use a calculator or your knowledge of the exact function values to write
one value of the variable to an acceptable degree of accuracy.
3. If the measure of the angle found in step 2 is not that of a quadrantal angle,
find the measure of its reference angle.
4. Use the measure of the reference angle to find the degree measures of each
solution in the interval 0° u , 360° or the radian measures of each solution in the interval 0 u , 2p.
5. Add 360n (n an integer) to the solutions in degrees found in steps 2 and 4 to
write all possible solutions in degrees.Add 2pn (n an integer) to the solutions
in radians found in steps 2 and 4 to write all possible solutions in radians.
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521
The following table will help you find the locations of the angles that satisfy
trigonometric equations. The values in the table follow from the definitions of
the trigonometric functions on the unit circle.
Sign of a and b (0 * |a| * 1, b 0)
1
2
sin u 5 a
Quadrants I and II
Quadrants III and IV
cos u 5 a
Quadrants I and IV
Quadrants II and III
tan u 5 b
Quadrants I and III
Quadrants II and IV
EXAMPLE 1
Find the solution set of the equation 7 tan u 5 2 !3 1 tan u in the interval
0° u , 360°.
Solution
How to Proceed
(1) Solve the equation for tan u:
7 tan u 5 2 !3 1 tan u
6 tan u 5 2 !3
tan u 5 !3
3
(2) Since tan u is positive, u1 can be a
first-quadrant angle:
u1 5 30°
(3) Since u is a first-quadrant angle,
R 5 u:
R 5 30°
(4) Tangent is also positive in the third
quadrant. Therefore, there is a
third-quadrant angle such that
u2 5 180° 1 R
u2 5 180° 1 30° 5 210°
tan u 5 !3
3 . In the third quadrant,
u2 5 180° 1 R:
Answer The solution set is {30°, 210°}.
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EXAMPLE 2
Find, to the nearest hundredth, all possible solutions of the following equation
in radians:
3(sin A 1 2) 5 3 2 sin A
Solution
How to Proceed
(1) Solve the equation for sin A:
3(sin A 1 2) 5 3 2 sin A
3 sin A 1 6 5 3 2 sin A
4 sin A 5 23
sin A 5 234
(2) Use a calculator to find one
value of A (be sure that the
calculator is in RADIAN
mode):
ENTER:
2nd
SIN1
(-)
3 4
)
ENTER
DISPLAY:
sin-1(-3/4)
-.848062079
One value of A is 20.848.
(3) Find the reference angle:
R 5 2A 5 2(20.848) 5 0.848
(4) Sine is negative in quadrants
III and IV. Use the
reference angle to find a
value of A in each of these
quadrants:
In quadrant III: A1 5 p 1 0.848 3.99
In quadrant IV: A2 5 2p 2 0.848 5.44
(5) Write the solution set:
{3.99 1 2pn, 5.44 1 2pn} Answer
Note: When sin21 A 234 B was entered, the calculator returned the value in the
interval 2p2 # u # p2 , the range of the inverse of the sine function. This is the
measure of a fourth-quadrant angle that is a solution of the equation. However, solutions are usually given as angle measures in radians between 0 and
2p plus multiples of 2p. Note that the value returned by the calculator is
5.43 1 2p (21) 20.85.
EXAMPLE 3
Find all possible solutions to the following equation in degrees:
1
2
( sec u 1 3) 5 sec u 1 52
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First-Degree Trigonometric Equations
Solution (1) Solve the equation for sec u:
1
2
( sec u 1 3) 5 sec u 1 52
sec u 1 3 5 2 sec u 1 5
22 5 sec u
(2) Rewrite the equation in terms
of cos u:
cos u 5 212
(3) Use a calculator to find one
value of u:
ENTER:
2nd
COS1 21
2
)
ENTER
DISPLAY:
cos-1(-1/2)
120
Cosine is negative in quadrant II so
u1 5 120°.
(4) Find the reference angle:
R 5 180 2 120 5 60°
(5) Cosine is also negative in
quadrant III. Use the reference
angle to find a value of u2 in
quadrant III:
In quadrant III: u2 5 180 1 60 5 240°
(6) Write the solution set:
{120 1 360n, 240 1 360n} Answer
Trigonometric Equations and the Graphing
Calculator
Just as we used the graphing calculator to approximate the irrational solutions
of quadratic-linear systems in Chapter 5, we can use the graphing calculator to
approximate the irrational solutions of trigonometric equations. For instance,
3(sin A 1 2) 5 3 2 sin A
from Example 2 can be solved using the intersect feature of the calculator.
STEP 1. Treat each side of the equation as a function.
Enter Y1 5 3(sin X 1 2) and Y2 5 3 2 sin X into the Y menu.
STEP 2. Using the following viewing window:
Xmin 5 0, Xmax 5 2p, Xscl 5 p6 ,
Ymin 5 0, Ymax 5 10
and with the calculator set to radian
mode, GRAPH the functions.
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STEP 3. The solutions are the x-coordinates of the intersection points of
the graphs. We can find the intersection points by using the intersect
function. Press 2nd CALC 5 ENTER ENTER to select both curves.
When the calculator asks you for a guess, move the cursor near one of
the intersection points using the arrow keys and then press ENTER .
Repeat this process to find the other intersection point.
*
*
Intersection
X=3.9896547 Y=3.75
Intersection
X=5.4351232 Y=3.75
As before, the solutions in the interval 0 u , 2p are approximately 3.99 and
5.44. The solution set is {3.99 1 2pn, 5.44 1 2pn}.
Exercises
Writing About Mathematics
1. Explain why the solution set of the equation 2x 1 4 5 8 is {2} but the solution set of the
equation 2 sin x 1 4 5 8 is { }, the empty set.
2. Explain why 2x 1 4 5 8 has only one solution in the set of real numbers but the equation
2 tan x 1 4 5 8 has infinitely many solutions in the set of real numbers.
Developing Skills
In 3–8, find the exact solution set of each equation if 0° u , 360°.
3. 2 cos u 2 1 5 0
4. 3 tan u 1 !3 5 0
5. 4 sin u 2 1 5 2 sin u 1 1
6. 5(cos u 1 1) 5 5
7. 3(tan u 2 2) 5 2 tan u 2 7
8. sec u 1 !2 5 2 !2
In 9–14, find the exact values for u in the interval 0 u , 2p.
9. 3 sin u 2 !35 sin u
10. 5 cos u 1 3 5 3 cos u 1 5
11. tan u 1 12 5 2 tan u 1 11
12. sin u 1 !2 5 !2
2
13. 3 csc u 1 5 5 csc u 1 9
14. 4(cot u 1 1) 5 2(cot u 1 2)
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525
In 15–20, find, to the nearest degree, the measure of an acute angle for which the given equation is
true.
15. sin u 1 3 5 5 sin u
16. 3 tan u 2 1 5 tan u 1 9
17. 5 cos u 1 1 5 8 cos u
18. 4(sin u 1 1) 5 6 2 sin u
19. csc u 2 1 5 3 csc u 2 11
20. cot u 1 8 5 3 cot u 1 2
In 21–24, find, to the nearest tenth, the degree measures of all u in the interval 0° u , 360° that
make the equation true.
21. 8 cos u 5 3 2 4 cos u
22. 5 sin u 2 1 5 1 2 2 sin u
23. tan u 2 4 5 3 tan u 1 4
24. 2 2 sec u 5 5 1 sec u
In 25–28, find, to the nearest hundredth, the radian measures of all u in the interval 0 u , 2p that
make the equation true.
25. 10 sin u 1 1 5 3 2 2 sin u
26. 9 2 2 cos u 5 8 2 4 cos u
27. 15 tan u 2 7 5 5 tan u 2 3
28. cot u 2 6 5 2 cot u 1 2
Applying Skills
29. The voltage E (in volts) in an electrical circuit is given by the function
E 5 20 cos (pt)
where t is time in seconds.
a. Graph the voltage E in the interval 0 t 2.
b. What is the voltage of the electrical circuit when t 5 1?
c. How many times does the voltage equal 12 volts in the first two seconds?
d. Find, to the nearest hundredth of a second, the times in the first two seconds when the
voltage is equal to 12 volts.
(1) Let u 5 pt. Solve the equation 20 cos u 5 12 in the interval 0 u , 2p.
(2) Use the formula u 5 pt and your answers to part (1) to find t when 0 u , 2p and
the voltage is equal to 12 volts.
30. A water balloon leaves the air
cannon at an angle of u with the
ground and an initial velocity of
40 feet per second. The water balloon lands 30 feet from the cannon. The distance d traveled by
the water balloon is given by the
formula
1 2
v sin 2u
d 5 32
where v is the initial velocity.
θ
30 ft
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1
a. Let x 5 2u. Solve the equation 30 5 32
(40) 2 sin x to the nearest tenth of a degree.
b. Use the formula x 5 2u and your answer to part a to find the measure of the angle
that the cannon makes with the ground.
31. It is important to understand the underlying mathematics before using the calculator to
solve trigonometric equations. For example, Adrian tried to use the intersect feature of his
graphing calculator to find the solutions of the equation cot u 5 sin A u 2 p2 B in the interval
0 u p but got an error message. Follow the steps that Adrian used to solve the
equation:
(1) Enter Y1 5 tan1 X and Y2 5 sin A X 2 p2 B into the Y menu.
(2) Use the following viewing window to graph the equations:
Xmin 5 0, Xmax 5 p, Xscl 5 p6 , Ymin 5 25, Ymax 5 5
(3) The curves seem to intersect at A p2 , 0 B . Press
2nd
CALC 5 ENTER
ENTER
to select both curves. When the calculator asks for a guess, move the cursor
near the intersection point using the arrow keys and then press ENTER .
a. Why does the calculator return an error message?
b. Is u 5 p2 a solution to the equation? Explain.
13-2 USING FACTORING TO SOLVE TRIGONOMETRIC EQUATIONS
We know that the equation 3x2 2 5x 2 2 5 0 can be solved by factoring the left
side and setting each factor equal to 0. The equation 3 tan2 u 2 5 tan u 2 2 5 0
can be solved for tan u in a similar way.
3x2 2 5x 2 2 5 0
3 tan2 u 2 5 tan u 2 2 5 0
(3x 1 1)(x 2 2) 5 0
(3 tan u 1 1)(tan u 2 2) 5 0
3x 1 1 5 0
3x 5 21
x5
213
x2250
x52
3 tan u 1 1 5 0
3 tan u 5 21
tan u 5
tan u 2 2 5 0
tan u 5 2
213
In the solution of the algebraic equation, the solution is complete. The solution set is U 213, 2 V . In the solution of the trigonometric equation, we must now
find the values of u.
There are two values of u in the interval 0° u , 360° for which tan u 5 2,
one in the first quadrant and one in the third quadrant. The calculator will display the measure of the first-quadrant angle, which is also the reference angle
for the third-quadrant angle.
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Using Factoring to Solve Trigonometric Equations
ENTER:
2nd
TAN1 2
)
ENTER
DISPLAY:
527
tan-1(2)
63.43494882
To the nearest tenth of a degree, the measure of u in the first quadrant is
63.4. This is also the reference angle for the third-quadrant angle.
In quadrant I: u1 5 63.4°
In quadrant III: u2 5 180 1 R 5 180 1 63.4 5 243.4°
There are two values of u in the interval 0° u , 360° for which tan u 5 213,
one in the second quadrant and one in the fourth quadrant.
ENTER:
2nd
TAN1 21
3
)
DISPLAY:
ENTER
tan-1(-1/3)
-18.43494882
The calculator will display the measure of a fourth-quadrant angle, which is
negative. To the nearest tenth of a degree, one measure of u in the fourth quadrant is 218.4. The opposite of this measure, 18.4, is the measure of the reference
angle for the second- and fourth-quadrant angles.
In quadrant II: u3 5 180 2 18.4 5 161.6°
In quadrant IV: u4 5 360 2 18.4 5 341.6°
The solution set of 3 tan2 u 2 5 tan u 2 2 5 0 is
{63.4°, 161.6°, 243.4°, 341.6°}
when 0° u , 360°.
EXAMPLE 1
Find all values of u in the interval 0 u , 2p for which 2 sin u 2 1 5 sin3 u.
Solution
How to Proceed
(1) Multiply both sides of the
equation by sin u:
(2) Write an equivalent equation with
0 as the right side:
(3) Factor the left side:
(4) Set each factor equal to 0 and
solve for sin u:
2 sin u 2 1 5 sin3 u
2 sin2 u 2 sin u 5 3
2 sin2 u 2 sin u 2 3 5 0
(2 sin u 2 3)(sin u 1 1) 5 0
2 sin u 2 3 5 0
2 sin u 5 3
sin u 1 1 5 0
sin u 5 21
sin u 5 32
(5) Find all possible values of u:
Answer u 5 3p
2
There is no value of u such that
sin u . 1. For sin u 5 21, u 5 3p
2.
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EXAMPLE 2
Find the solution set of 4 sin2 A 2 1 5 0 for the degree measures of A in the
interval 0° A , 360°.
Solution
METHOD 1
METHOD 2
Solve for sin2 A and take the square
root of each side of the equation.
Factor the left side.
4 sin2 A 2 1 5 0
4 sin2 A 2 1 5 0
(2 sin A 2 1)(2 sin A 1 1) 5 0
2 sin A 2 1 5 0
4 sin2 A 5 1
2 sin A 1 1 5 0
2 sin A 5 1
2 sin A 5 21
sin A 5 12
sin A 5 212
sin2 A 5 14
sin A 5 612
If sin A 5 21, A 5 30° or A 5 150°. If sin A 5 212, A 5 210° or A 5 330°.
Answer {30°, 150°, 210°, 330°}
Note: The graphing calculator does not use the notation sin2 A, so we must enter
the square of the trig function as (sin A)2 or enter sin (A)2. For example, to
check the solution A 5 30° for Example 2:
ENTER: 4
(
4 SIN
DISPLAY:
SIN
30
30
)
)
)
x2
x2
1 ENTER
1 ENTER
4(sin(30))2-1
0
4sin(30)2-1
0
Factoring Equations with Two Trigonometric
Functions
To solve an equation such as 2 sin u cos u 1 sin u 5 0, it is convenient to rewrite
the left side so that we can solve the equation with just one trigonometric function value. In this equation, we can rewrite the left side as the product of two
factors. Each factor contains one function.
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529
2 sin u cos u 1 sin u 5 0
sin u (2 cos u 1 1) 5 0
sin u 5 0
2 cos u 1 1 5 0
u1 5 0°
2 cos u 5 21
cos u 5 212
u2 5 180°
Since cos 60° 5 12, R 5 60°.
In quadrant II: u3 5 180 2 60 5 120°
In quadrant III: u4 5 180 1 60 5 240°
The solution set is {0°, 120°, 180°, 240°}.
EXAMPLE 3
Find, in radians, all values of u in the interval 0 u 2p that are in the solution
set of:
sec u csc u 1 !2 csc u 5 0
Solution Factor the left side of the equation and set each factor equal to 0.
sec u csc u 1 !2 csc u 5 0
csc u Asec u 1 !2B 5 0
csc u 5 0 ✘
No solution
sec u 1 !2 5 0
sec u 5 2 !2
1
5 2 !2
cos u 5 2 !2
2
p
Since cos p4 5 !2
2 , R 5 4.
In quadrant II: u 5 p 2 p4 5 3p
4
In quadrant III: u 5 p 1 p4 5 5p
4
Answer
5p
U 3p
4, 4 V
Exercises
Writing About Mathematics
1. Can the equation tan u 1 sin u tan u 5 1 be solved by factoring the left side of the equation? Explain why or why not.
2. Can the equation 2(sin u)(cos u) 1 sin u 1 2 cos u 1 1 5 0 be solved by factoring the left
side of the equation? Explain why or why not.
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Developing Skills
In 3–8, find the exact solution set of each equation if 0° u , 360°.
3. 2 sin2 u 1 sin u 2 1 5 0
4. 3 tan2 u 5 1
5. tan2 u 2 3 5 0
6. 2 sin2 u 2 1 5 0
7. 6 cos2 u 1 5 cos u 2 4 5 0
8. 2 sin u cos u 1 cos u 5 0
In 9–14, find, to the nearest tenth of a degree, the values of u in the interval 0° u , 360° that satisfy each equation.
9. tan2 u 2 3 tan u 1 2 5 0
10. 3 cos2 u 2 4 cos u 1 1 5 0
11. 9 sin2 u 2 9 sin u 1 2 5 0
12. 25 cos2 u 2 4 5 0
13. tan2 u 1 4 tan u 2 12 5 0
14. sec2 u 2 7 sec u 1 12 5 0
In 15–20, find, to the nearest hundredth of a radian, the values of u in the interval 0 u , 2p that
satisfy the equation.
15. tan2 u 2 5 tan u 1 6 5 0
16. 4 cos2 u 2 3 cos u 5 1
17. 5 sin2 u 1 2 sin u 5 0
18. 3 sin2 u 1 7 sin u 1 2 5 0
19. csc2 u 2 6 csc u 1 8 5 0
20. 2 cot2 u 2 13 cot u 1 6 5 0
21. Find the smallest positive value of u such that 4 sin2 u 2 1 5 0.
22. Find, to the nearest hundredth of a radian, the value of u such that sec u 5 sec5 u and
p
2 , u , p.
23. Find two values of A such that (sin A)(csc A) 5 2sin A.
13-3 USING THE QUADRATIC FORMULA TO SOLVE
TRIGONOMETRIC EQUATIONS
Not all quadratic equations can be solved by factoring. It is often useful or necessary to use the quadratic formula to solve a second-degree trigonometric
equation.
The trigonometric equation 2 cos2 u 2 4 cos u 1 1 5 0 is similar in form to
the algebraic equation 2x2 2 4x 1 1 5 0. Both are quadratic equations that cannot be solved by factoring over the set of integers but can be solved by using the
quadratic formula with a 5 2, b 5 24, and c 5 1.
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Algebraic equation:
Trigonometric equation:
2x2 2 4x 1 1 5 0
2 cos2 u 2 4 cos u + 1 5 0
2
x 5 2b 6 "2ab 2 4ac
2
cos u 5 2b 6 "2ab 2 4ac
x5
cos u 5
x5
x5
x5
531
2(24) 6 " (24) 2 2 4(2)(1)
2(2)
4 6 !16 2 8
4
4 6 !8
4
2 6 !2
2
cos u 5
cos u 5
cos u 5
2(24) 6 " (24) 2 2 4(2)(1)
2(2)
4 6 !16 2 8
4
4 6 !8
4
2 6 !2
2
There are no differences between the two solutions up to this point.
However, for the algebraic equation, the solution is complete. There are two val-
ues of x that make the equation true: x 5 2 12 !2 or x 5 2 22 !2.
For the trigonometric equation, there appear to be two values of cos u. Can
we find values of u for each of these two values of cos u?
CASE 1
cos u 5 2 12 !2 1 1 21.414 5 1.207
There is no value of u such that cos u . 1.
CASE 2
cos u 5 2 22 "2 2 2 21.414 5 0.293
There are values of u in the first quadrant and in the fourth quadrant such
that cos u is a positive number less than 1. Use a calculator to approximate these
values to the nearest degree.
ENTER:
2nd
COS1
(
2nd
¯ 2
)
2
)
2 DISPLAY:
cos-1((2-√(2))/2)
72.96875154
)
ENTER
To the nearest degree, the value of u in the first quadrant is 73°. This is
also the value of the reference angle. Therefore, in the fourth quadrant,
u 5 360° 2 73° or 287°.
In the interval 0° u , 360°, the solution set of 2 cos2 u 2 4 cos u 1 1 5 0 is:
{73°, 287°}
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EXAMPLE 1
a. Use three different methods to solve tan2 u 2 1 5 0 for tan u.
b. Find all possible values of u in the interval 0 u , 2p.
Solution a.
METHOD 1: FACTOR
METHOD 2: SQUARE ROOT
2
tan u 2 1 5 0
tan 2 u 2 1 5 0
(tan u 1 1)(tan u 2 1) 5 0
tan2 u 5 1
tan u 1 1 5 0
tan u 2 1 5 0
tan u 5 21
tan u 5 61
tan u 5 1
METHOD 3: QUADRATIC FORMULA
tan2 u 2 1 5 0
a 5 1, b 5 0, c 5 21
tan u 5
20 6 "02 2 4(1)(21)
2(1)
b. When tan u 5 1, u is in quadrant I or
in quadrant III.
In quadrant I: if tan u 5 1, u 5 p4
This is also the measure of the
reference angle.
In quadrant III: u 5 p 1 p4 5 5p
4
Answer
5 6 2!4 5 62
2 5 61
When tan u 5 21, u is in quadrant
II or in quadrant IV.
One value of u is 2p4 . The reference angle is p4 .
In quadrant II: u 5 p 2 p4 5 3p
4
In quadrant IV: u 5 2p 2 p4 5 7p
4
3p 5p 7p
Up
4, 4 , 4 , 4 V
EXAMPLE 2
Find, to the nearest degree, all possible values of B such that:
3 sin2 B 1 3 sin B 2 2 5 0
a 5 3, b 5 3, c 5 22
Solution
sin B 5
CASE 1
23 6 " (3) 2 2 4(3)(22)
2(3)
1 24
5 23 66 !33
5 23 6 !9
6
Let sin B 5 23 16 !33.
Since 23 16 !33 < 0.46 is a number between 21 and 1, it is in the range of the
sine function.
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Using the Quadratic Formula to Solve Trigonometric Equations
ENTER:
2nd
SIN1
2nd
¯ 33
6
)
23 (
)
DISPLAY:
)
sin-1((-3+√(33))
/6)
27.22120768
ENTER
The sine function is positive in the first and second quadrants.
In quadrant I: B 5 27°
In quadrant II: B 5 180 2 27 5 153°
CASE 2
Let sin B 5 23 26 !33.
Since 23 26 !33 < 21.46 is not a number between 21 and 1, it is not in
the range of the sine function. There are no values of B such that
sin B 5 23 26 !33. ✘
Calculator Enter Y1 5 3 sin2 X 1 3 sin X 2 2 into the Y menu.
Solution
)
3 SIN
X,T,,n
x2
ENTER: Y 3 SIN
X,T,,n
)
2
With the calculator set to DEGREE mode, graph
the function in the following viewing window:
Xmin 5 0, Xmax 5 360, Xscl 5 30, Ymin 5 25,
Ymax 5 5
The solutions are the x-coordinates of the xintercepts, that is, the roots of Y1. Use the zero
function of your graphing calculator to find the
roots. Press 2nd CALC 2 . Use the arrows
to enter a left bound to the left of one of the zero values, a right bound to the
right of the zero value, and a guess near the zero value. The calculator will display the coordinates of the point at which the graph intersects the x-axis.
Repeat to find the other root.
*
*
Zero
X=27.221208
Y=0
Zero
X=152.77879
Y=0
As before, we find that the solutions in the interval 0° u , 360° are approximately 27° and 153°.
Answer B 5 27° 1 360n or B 5 153° 1 360n for integral values of n.
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Trigonometric Equations
Exercises
Writing About Mathematics
1. The discriminant of the quadratic equation tan2 u 1 4 tan u 1 5 5 0 is 24. Explain why the
solution set of this equation is the empty set.
2. Explain why the solution set of 2 csc2 u 2 csc u 5 0 is the empty set.
Developing Skills
In 3–14, use the quadratic formula to find, to the nearest degree, all values of u in the interval
0° u , 360° that satisfy each equation.
3. 3 sin2 u 2 7 sin u 2 3 5 0
4. tan2 u 2 2 tan u 2 5 5 0
5. 7 cos2 u 2 1 5 5 cos u
6. 9 sin2 u 1 6 sin u 5 2
7. tan2 u 1 3 tan u 1 1 5 0
8. 8 cos2 u 2 7 cos u 1 1 5 0
9. 2 cot2 u 1 3 cot u 2 4 5 0
10. sec2 u 2 2 sec u 2 4 5 0
11. 3 csc2 u 2 2 csc u 5 2
12. 2 tan u (tan u 1 1) 5 3
13. 3 cos u 1 1 5 cos1 u
14. sin2 u 5 sin u3 1 2
1
15. Find all radian values of u in the interval 0 u , 2p for which sin1 u 5 2 sin
u.
16. Find, to the nearest hundredth of a radian, all values of u in the interval 0 u , 2p for
which cos3 u 5 3 cos 1u 1 1.
13-4 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC
EQUATIONS INVOLVING MORE THAN ONE FUNCTION
When an equation contains two different functions, it may be possible to factor
in order to write two equations, each with a different function. We can also use
identities to write an equivalent equation with one function.
The equation cos2 u 1 sin u 5 1 cannot be solved by factoring. We can use
the identity cos2 u 1 sin2 u 5 1 to change the equation to an equivalent equation in sin u by replacing cos2 u with 1 2 sin2 u.
cos2 u 1 sin u 5 1
1 2 sin2 u 1 sin u 5 1
2sin2 u 1 sin u 5 0
sin u (2sin u 1 1) 5 0
sin u 5 0
2sin u 1 1 5 0
u15 0°
1 5 sin u
u2 5 180°
u3 5 90°
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Using Substitution to Solve Trigonometric Equations Involving More than One Function
Check u1 5 0°
cos 2 u 1 sin u 5 1
Check u2 5 180°
Check u3 5 90°
cos 2 u 1 sin u 5 1
cos 2 u 1 sin u 5 1
?
cos 2 1808 1 sin 1808 5 1
?
(21) 2 1 0 5 1
cos 2 08 1 sin 08 5 1
12 1 0 5 1
151✔
?
cos 2 908 1 sin 908 5 1
?
?
02 1 1 5 1
?
151✔
151✔
In the interval 0° u , 360°, the solution set of cos u 1 sin u 5 1 is
{0°, 90°, 180°}.
2
Is it possible to solve the equation cos2 u 1 sin u 5 1 by writing an equivalent equation in terms of cos u? To do so we must use an identity to write sin u
in terms of cos u. Since sin2 u 5 1 2 cos2 u, sin u 5 6"1 2 cos2 u.
cos2 u 1 sin u 5 1
(1) Write the equation:
cos2 u 6"1 2 cos2 u 5 1
(2) Replace sin u with
6"1 2 cos2 u:
6"1 2 cos2 u 5 1 2 cos2 u
(3) Isolate the radical:
1 2 cos2 u 5 1 2 2 cos2 u 1 cos4 u
(4) Square both sides of
the equation:
(5) Write an equivalent
equation with the right
side equal to 0:
cos2 u 2 cos4 u 5 0
cos2 u (1 2 cos2 u) 5 0
(6) Factor the left side:
(7) Set each factor equal to 0:
cos2 u 5 0
cos u 5 0
u15 90°
u2 5 270°
1 2 cos2 u 5 0
1 5 cos2 u
61 5 cos u
u3 5 0°
u4 5 180°
This approach uses more steps than the first. In addition, because it involves
squaring both sides of the equation, an extraneous root, 270°, has been introduced. Note that 270° is a root of the equation cos2 u 2 cos4 u 5 0 but is not a
root of the given equation.
cos2 u 1 sin u 5 1
?
cos2 270° 1 sin 270° 5 1
?
(0)2 1 (21) 5 1
0211✘
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Any of the eight basic identities or the related identities can be substituted
in a given equation.
EXAMPLE 1
Find all values of A in the interval 0° A , 360° such that 2 sin A 1 1 5 csc A.
Solution
How to Proceed
(1) Write the equation:
2 sin A 1 1 5 csc A
(2) Replace csc A with sin1 A:
2 sin A 1 1 5 sin1 A
(3) Multiply both sides of the
equation by sin A:
(4) Write an equivalent equation
with 0 as the right side:
(5) Factor the left side:
(6) Set each factor equal to 0
and solve for sin A:
2 sin2 A 1 sin A 5 1
2 sin2 A 1 sin A 2 1 5 0
(2 sin A 2 1)(sin A 1 1) 5 0
2 sin A 2 1 5 0
sin A 1 1 5 0
2 sin A 5 1
sin A 5 21
sin A 5 12
A 5 270°
A 5 30°
or A 5 150°
Answer A 5 30° or A 5 150° or A 5 270°
Often, more than one substitution is necessary to solve an equation.
EXAMPLE 2
If 0 u , 2p, find the solution set of the equation 2 sin u 5 3 cot u.
Solution
How to Proceed
(1) Write the equation:
u
(2) Replace cot u with cos
sin u :
(3) Multiply both sides of the
equation by sin u:
2 sin u 5 3 cot u
u
2 sin u 5 3 A cos
sin u B
2 sin2 u 5 3 cos u
(4) Replace sin2 u with 1 2 cos2 u:
2(1 2 cos2 u) 5 3 cos u
(5) Write an equivalent equation
in standard form:
2 2 2 cos2 u 5 3 cos u
2 2 2 cos2 u 2 3 cos u 5 0
2 cos2 u 1 3 cos u 2 2 5 0
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Using Substitution to Solve Trigonometric Equations Involving More than One Function
(6) Factor and solve for cos u:
(2 cos u 2 1)(cos u 1 2) 5 0
(7) Find all values of u in the
given interval:
2 cos u 2 1 5 0
cos u 1 2 5 0
cos u 5 22 ✘
2 cos u 5 1
cos u 5
u5
or u 5
Answer
537
1
2
p
3
5p
3
No solution
5p
Up
3, 3 V
The following identities from Chapter 10 will be useful in solving trigonometric equations:
Reciprocal Identities
Quotient Identities
Pythagorean Identities
sec u 5cos1 u
sin u
tan u 5cos
u
cos2 u 1 sin2 u 5 1
csc u 5sin1 u
u
cot u 5cos
sin u
1 1 tan2 u 5 sec2 u
cot u 5tan1 u
cot2 u 1 1 5 csc2 u
Exercises
Writing About Mathematics
1. Sasha said that sin u 1 cos u 5 2 has no solution. Do you agree with Sasha? Explain why or
why not.
2. For what values of u is sin u 5 "1 2 cos2 u true?
Developing Skills
In 3–14, find the exact values of u in the interval 0° u , 360° that satisfy each equation.
3. 2 cos2 u 2 3 sin u 5 0
4. 4 cos2 u 1 4 sin u 2 5 5 0
5. csc2 u 2 cot u 2 1 5 0
6. 2 sin u 1 1 5 csc u
2
7. 2 sin u 1 3 cos u 2 3 5 0
9. 2 cos u 5 sec u
8. 3 tan u 5 cot u
10. sin u 5 csc u
11. tan u 5 cot u
12. 2 cos2 u 5 sin u 1 2
13. cot2 u 5 csc u 1 1
14. 2 sin2 u 2 tan u cot u 5 0
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Applying Skills
15. An engineer would like to model a piece for a factory
machine on his computer. As shown in the figure, the
machine consists of a link fixed to a circle at point A.
The other end of the link is fixed to a slider at point
B. As the circle rotates, point B slides back and forth
between the two ends of the slider (C and D). The
movement is restricted so that u, the measure of
AOD, is in the interval 245° u 45°. The motion
of point B can be described mathematically by the
formula
A
u
O
C
B
D
CB 5 r(cos u 2 1) 1 "l 2 2 r 2 sin2 u
where r is the radius of the circle and l is the length of the link. Both the radius of the circle
and the length of the link are 2 inches.
a. Find the exact value of CB when: (1) u 5 30° (2) u 5 45°.
b. Find the exact value(s) of u when CB 5 2 inches.
c. Find, to the nearest hundredth of a degree, the value(s) of u when CB 5 1.5 inches.
13-5 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC
EQUATIONS INVOLVING DIFFERENT ANGLE MEASURES
If an equation contains function values of two different but related angle measures, we can use identities to write an equivalent equation in terms of just one
variable. For example: Find the value(s) of u such that sin 2u 2 sin u 5 0.
Recall that sin 2u 5 2 sin u cos u. We can use this identity to write the equation in terms of just one variable, u, and then use any convenient method to
solve the equation.
(1) Write the equation:
sin 2u 2 sin u 5 0
(2) For sin 2u, substitute its equal,
2 sin u cos u:
2 sin u cos u 2 sin u 5 0
(3) Factor the left side:
(4) Set each factor equal to zero
and solve for u:
sin u (2 cos u 2 1) 5 0
sin u 5 0
u 5 0°
or u 5 180°
2 cos u 2 1 5 u
2 cos u 5 1
cos u 5 12
u 5 60°
or u 5 300°
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539
The degree measures 0°, 60°, 180°, and 300° are all of the values in the interval 0° u , 360° that make the equation true. Any values that differ from these
values by a multiple of 360° will also make the equation true.
EXAMPLE 1
Find, to the nearest degree, the roots of cos 2u 2 2 cos u 5 0.
Solution
How to Proceed
(1) Write the given equation:
cos 2u 2 2 cos u 5 0
(2) Use an identity to write
cos 2u in terms of cos u:
2 cos2 u 2 1 2 2 cos u 5 0
(3) Write the equation in standard
form:
2 cos2 u 2 2 cos u 2 1 5 0
(4) The equation cannot be factored
over the set of integers. Use the
quadratic formula:
cos u 5
2(22) 6 " (22) 2 2 4(2)(21)
2(2)
5 2 6 4!12
5 1 62 !3
(5) When we use a calculator to
approximate the value of
arccos 1 22 !3, the calculator will
return the value 111°. Cosine is
negative in both the second and
the third quadrants. Therefore,
there is both a second-quadrant
and a third-quadrant angle such
that cos u 5 1 22 !3:
(6) When we use a calculator to
approximate the value of
arccos 1 12 !3, the calculator will
return an error message because
1 1 !3
2
. 1 is not in the domain
of arccosine:
Answer To the nearest degree, u 5 111° or u 5 249°.
cos-1((1-√(3))/2)
111.4707014
u1 5 111°
R 5 180° 2 111° 5 69°
u2 5 180° 1 69° 5 249°
ERR:DOMAIN
1: Quit
2:Goto
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EXAMPLE 2
Find, to the nearest degree, the values of u in the interval 0° u 360° that are
solutions of the equation sin (90° 2 u) 1 2 cos u 5 2.
Solution Use the identity sin (90° 2 u) 5 cos u.
sin(90° 2 u) 1 2 cos u 5 2
cos u 1 2 cos u 5 2
3 cos u 5 2
cos u 5 32
To the nearest degree, a calculator returns the value of u as 48°.
In quadrant I, u 5 48° and in quadrant IV, u 5 360° 2 48° 5 312°.
Answer u 5 48° or u 5 312°
The basic trigonometric identities along with the cofunction, double-angle,
and half-angle identities will be useful in solving trigonometric equations:
Cofunction Identities
Double-Angle Identities
Half-Angle Identities
cos u 5 sin (90° 2 u)
sin (2u) 5 2 sin u cos u
sin 12u 5 6#1 2 2cos u
sin u 5 cos (90° 2 u)
cos (2u) 5 cos2 u 2 sin2 u
tan u 5 cot (90° 2 u)
2 tan u
tan (2u) 5 1 2
tan2 u
cot u 5 tan (90° 2 u)
cos 12u 5 6#1 1 2cos u
2 cos u
tan 12u 5 6#11 1
cos u
sec u 5 csc (90° 2 u)
csc u 5 sec (90° 2 u)
Note that in radians, the right sides of the cofunction identities are written
in terms of p2 2 u.
Exercises
Writing About Mathematics
1. Isaiah said that if the equation cos 2x 1 2 cos2 x 5 2 is divided by 2, an equivalent equation
is cos x 1 cos2 x 5 1. Do you agree with Isaiah? Explain why or why not.
2. Aaron solved the equation 2 sin u cos u 5 cos u by first dividing both sides of the equation
by cos u. Aaron said that for 0 u 2p, the solution set is U p6 , 5p
6 V . Do you agree with
Aaron? Explain why or why not.
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Using Substitution to Solve Trigonometric Equations Involving Different Angle Measures
Developing Skills
In 3–10, find the exact values of u in the interval 0° u 360° that make each equation true.
3. sin 2u 2 cos u 5 0
4. cos 2u 1 sin2 u 5 1
5. sin 2u 1 2 sin u 5 0
6. tan 2u 5 cot u
2
7. cos 2u 1 2 cos u 5 2
9. 3 2 3 sin u 2 2 cos2 u 5 0
8. sin 12u 5 cos u
10. 3 cos 2u 2 4 cos2 u 1 2 5 0
In 11–18, find all radian measures of u in the interval 0 u 2p that make each equation true.
Express your answers in terms of p when possible; otherwise, to the nearest hundredth.
11. cos 2u 5 2 cos u 2 2 cos2 u
12. 2 sin 2u 1 sin u 5 0
13. 5 sin2 u 2 4 sin u 1 cos 2u 5 0
14. cos u 5 3 sin 2u
15. 3 sin 2u 5 tan u
17. 2 cos2 u 1 3 sin u 2 2 cos 2u 5 1
16. sin A p2 2 u B 1 cos2 u 5 14
18. (2 sin u cos u)2 1 4 sin 2u 2 1 5 0
Applying Skills
19. Martha swims 90 meters from point A on the
north bank of a stream to point B on the opposite
bank. Then she makes a right angle turn and
swims 60 meters from point B to point C, another
point on the north bank. If mCAB 5 u, then
mACB 5 90° 2 u.
C
A
u
90° 2 u
d
a. Let d be the width of the stream, the length of
the perpendicular distance from B to AC.
Express d in terms of sin u.
B
b. Express d in terms of sin (90° 2 u).
c. Use the answers to a and b to write an equation. Solve the equation for u.
d. Find d, the width of the stream.
20. A pole is braced by two wires of equal length as
shown in the diagram. One wire, AB, makes an angle
of u with the ground, and the other wire, CD, makes
an angle of 2u with the ground. If FD 5 1.75FB, find,
to the nearest degree, the measure of u:
D
B
a. Let AB = CD = x, FB 5 y, and FD 5 1.75y.
Express sin u and sin 2u in terms of x and y.
b. Write an equation that expresses a relationship
between sin u and sin 2u and solve for u to the
nearest degree.
2u
C
u
F
A
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CHAPTER SUMMARY
A trigonometric equation is an equation whose variable is expressed in
terms of a trigonometric function value.
The following table compares the degree measures of u from 290° to
360°, the radian measures of u from 2p2 to 2p, and the measure of its reference
angle.
Angle
Reference
Angle
Fourth
Quadrant
First
Quadrant
Second
Quadrant
Third
Quadrant
Fourth
Quadrant
290° , u , 0°
2p2 , u , 0
0° , u , 90°
0 , u , p2
90° , u , 180°
p
2 ,u,u
180° , u , 270°
p , u , 3p
2
270° , u , 360°
3p
2 , u , 2p
2u
2u
u
u
180° 2 u
p2u
u 2 180°
u2p
360° 2 u
2p 2 u
To solve a trigonometric equation:
1. If the equation involves more than one variable, use identities to write the
equation in terms of one variable.
2. If the equation involves more than one trigonometric function of the same
variable, separate the functions by factoring or use identities to write the
equation in terms of one function of one variable.
3. Solve the equation for the function value of the variable. Use factoring or
the quadratic formula to solve a second-degree equation.
4. Use a calculator or your knowledge of the exact function values to write
one value of the variable to an acceptable degree of accuracy.
5. If the measure of the angle found in step 4 is not that of a quadrantal
angle, find the measure of its reference angle.
6. Use the measure of the reference angle to find the degree measures of
each solution in the interval 0° u , 360° or the radian measures of each
solution in the interval 0 u , 2p.
7. Add 360n (n an integer) to the solutions in degrees found in steps 4 and 6
to write all possible solutions in degrees. Add 2pn (n an integer) to the
solutions in radians found in steps 2 and 4 to write all possible solutions in
radians.
VOCABULARY
13-1 Trigonometric equation
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Review Exercises
REVIEW EXERCISES
In 1–10, find the exact values of x in the interval 0° x 360° that make each
equation true.
1. 2 cos x 1 1 5 0
2. !3 2 sin x 5 sin x 1 !12
3. 2 sec x 5 2 1 sec x
4. 2 cos2 x 1 cos x 2 1 5 0
5. cos x sin x 1 sin x 5 0
6. tan x 2 3 cot x 5 0
7. 2 cos x 2 sec x 5 0
8. sin2 x 2 cos2 x 5 0
9. 2 tan x 5 1 2 tan2 x
10. cos3 x 2 34 cos x 5 0
In 11–22, find, to the nearest hundredth, all values of u in the interval
0 u 2p that make each equation true.
11. 7 sin u 1 3 5 1
12. 5(cos u 2 1) 5 6 1 cos u
13. 4 sin2 u 2 3 sin u 5 1
14. 3 cos2 u 2 cos u 2 2 5 0
15. tan2 u 2 4 tan u 2 1 5 0
16. sec2 u 2 10 sec u 1 20 5 0
17. tan 2u 5 4 tan u
18. 2 sin 2u 1 cos u 5 0
2u
19. 1 1sincos
2u 5 4
20. 3 cos 2u 1 cos u 1 2 5 0
21. 2 tan2 u 1 6 tan u 5 20
22. cos 2u 2 cos2 u 1 cos u 1 14 5 0
23. Explain why the solution set of tan u 2 sec u 5 0 is the empty set.
24. In ABC, mA 5 u and mB 5 2u. The
altitude from C intersects AB at D and
AD : DB 5 5 : 2.
a. Write tan u and tan 2u as ratios of the
sides of ADC and BDC, respectively.
C
u
A
D
2u B
b. Solve the equation for tan 2u found in
a for CD.
c. Substitute the value of CD found in b into the equation of tan u found
in a.
d. Solve for u.
e. Find the measures of the angles of the triangle to the nearest degree.
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Trigonometric Equations
Exploration
For (1)–(6): a. Use your knowledge of geometry and trigonometry to express
the area, A, of the shaded region in terms of u. b. Find the measure of u when
A 5 0.5 square unit or explain why there is no possible value of u. Give the exact
value when possible; otherwise, to the nearest hundredth of a radian.
(1)
(2)
u
2
2
u
0 , u , p2
0 , u , p2
(3)
(4)
1
1
u
1
O
1
1
2u
1
0 , u , p2
(5)
0,u,p
(6)
1
u
O
p
2
1
,u#p
u
1 O
0 , u , p2
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Cumulative Review
CUMULATIVE REVIEW
545
CHAPTERS 1–13
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1
1. The expression 2(5) 0 1 3(27) 23 is equal to
(2) 2 1 !3
9
(1) 227
2. The sum of 34a2 2 12a and a 2 12a2 is
(1) 14a2 1 12a
(3) 3
(4) 219
(3) 54a2 2 32a
(4) 74a2 2 a
(2) 2a3
3. The fraction 1 24 !5 is equivalent to
(1) 1 1 !5
(2) 1 2 !5
(3) 21 + !5
(4) 21 2 !5
4. The complex number i12 1 i10 can be written as
(1) 0
(2) 1
(3) 2
(4) 1 1 i
4
5. a S (21) n n2 T is equal to
n51
(1) 25
(2) 1
(3) 2
(4) 5
6. When the roots of a quadratic equation are real and irrational, the discriminant must be
(1) zero.
(2) a positive number that is a perfect square.
(3) a positive number that is not a perfect square.
(4) a negative number.
7. When f(x) 5 x2 1 1 and g(x) 5 2x, then g(f(x)) equals
(1) 4x2 1 1
(3) 4x2 1 2
(2) 2x2 1 2x 1 1
(4) 2x2 1 2
8. If log x 5 2 log a 2 13 log b, then x equals
(1) 2a 2 13b
2
(3) 1a
(2) a2 2 !b
(4)
3
3b
a2
3
!b
9. What number must be added to the binomial x2 1 5x in order to change it
into a trinomial that is a perfect square?
(1) 52
(2) 25
4
(3) 25
2
(4) 25
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Trigonometric Equations
10. If f(x) is a function from the set of real numbers to the set of real numbers, which of the following functions is one-to-one and onto?
(1) f(x) 5 2x 2 1
(3) f(x) 5 2x 2 1
2
(2) f(x) = x
(4) f(x) 5 2x2 1 2x
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Sketch the graph of the inequality y 2x2 2 2x 1 3.
12. When f(x) 5 4x 2 2, find f21(x), the inverse of f(x).
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Write the first six terms of the geometric function whose first term is 2 and
whose fourth term is 18.
14. The endpoints of a diameter of a circle are (22, 5) and (4, 21). Write an
equation of the circle.
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. If csc u 5 3, and cos u , 0, find sin u, cos u, tan u, cot u and sec u.
16. Solve for x and write the roots in a + bi form: x6 2 2 5 x52.
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CHAPTER
TRIGONOMETRIC
APPLICATIONS
An ocean is a vast expanse that can be life-threatening to a person who experiences a disaster while
boating. In order for help to arrive on time, it is necessary that the coast guard or a ship in the area be able
to make an exact identification of the location. A distress signal sent out by the person in trouble can be
analyzed by those receiving the signal from different
directions. In this chapter we will derive the formulas
that can be used to determine distances and angle measures when sufficient information is available.
14
CHAPTER
TABLE OF CONTENTS
14-1 Similar Triangles
14-2 Law of Cosines
14-3 Using the Law of Cosines to
Find Angle Measure
14-4 Area of a Triangle
14-5 Law of Sines
14-6 The Ambiguous Case
14-7 Solving Triangles
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
547
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14-1 SIMILAR TRIANGLES
In the study of geometry, we learned that if two triangles are similar, the corresponding angles are congruent and the corresponding sides are in proportion. If
certain pairs of corresponding angles and sides are congruent or proportional,
then the triangles must be similar. The following pairs of congruent corresponding angles and proportional corresponding sides are sufficient to prove
triangles similar.
1. Two angles (AA~)
2. Three sides (SSS~)
3. Two sides and the included angles (SAS~)
4. Hypotenuse and one leg of a right triangle (HL~)
We can use similar triangles to write the coordinates of a point in terms
of its distance from the origin and the sine and cosine of an angle in standard
position.
y
Let B(x1, y1) be a point in the first
B(x1, y1)
quadrant of the coordinate plane with
(0, 1)
OB 5 b and u the measure of the angle
P(cos u, sin u)
h
formed by OB and the positive ray of
u
the x-axis. Let P(cos u, sin u) be the point
O
A(x1, 0) x
Q
h
at which OB intersects the unit circle.
Let Q(cos u, 0) be the point at which a
perpendicular line from P intersects the
x-axis and A(x1, 0) be the point at which
a perpendicular line from B intersects
the x-axis. Then OPQ ~ OBA by
AA~. Therefore:
OA
OQ
x1 2 0
cos u
x1
cos u
OB
5 OP
b
51
b
51
x1 5 b cos u
AB
QP
y1 2 0
sin u
y1
sin u
OB
5 OP
b
51
b
51
y1 5 b sin u
Therefore, if B is a point b units from the origin of the coordinate plane and
h
OB is the terminal side of an angle in standard position whose measure is u, then
the coordinates of B are (b cos u, b sin u). This statement can be shown to be
true for any point in the coordinate plane.
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Similar Triangles
Let AOB be an angle in standard position whose measure is u. If the terh
minal side, OB, intersects the unit circle at P, then the coordinates of P are
(cos u, sin u). The diagrams below show AOB in each quadrant.
y
y
B
B
P(cos u, sin u)
P(cos u, sin u)
u
u
O
A
y
O
x
A
x
y
u
u
O
P(cos u, sin u)
A
x
O
x
A
P(cos u, sin u)
B
B
A dilation of b with center at the origin will stretch each segment whose
endpoint is the origin by a factor of b. Under the dilation Db, the image of (x, y)
is (bx, by). If B is the image of P(cos u, sin u) under the dilation Db, then the
coordinates of B are (b cos u, b sin u). Therefore, (b cos u, b sin u) are the coordinates of a point b units from the origin on the terminal ray of an angle in standard position whose measure is u.
Any triangle can be positioned on the coordinate plane so that each vertex
is identified by the coordinates of a point in the plane. The coordinates of the
vertices can be expressed in terms of the trigonometric function values of angles
in standard position.
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Trigonometric Applications
EXAMPLE 1
h
Point S is 12 units from the origin and OS makes an angle of 135° with the positive ray of the x-axis. What are the exact coordinates of S?
y
Solution The coordinates of S are
(12 cos 135°, 12 sin 135°).
S(12 cos 135°, 12 sin 135°)
Since 135° 5 180° 2 45°,
cos 135° 5 2cos 45° 5 2 !2
2 and
135°
sin 135° 5 sin 45° 5 !2
2 .
O
x
Therefore, the coordinates of S are
!2
Q12 3 2!2
2 , 12 3 2 R 5 A26 !2, 6 !2B . Answer
EXAMPLE 2
The coordinates of A are (25.30, 28.48).
a. Find OA to the nearest hundredth.
b. Find, to the nearest degree, the measure of the angle in standard position
h
whose terminal side is OA.
Solution a. Let C (25.30, 0) be the point at which a
vertical line from A intersects the x-axis.
Then OA is the hypotenuse of right OAC,
OC 5 25.30 2 0 5 5.30 and
AC 5 28.48 2 0 5 8.48.
Using the Pythagorean Theorem:
y
C(25.30, 0)
1 u
O1
x
OA2 5 OC 2 1 AC 2
OA2 5 5.302 1 8.482
OA 5 "5.302 1 8.482
ENTER:
¯
5.30 x 2
8.48 x 2
)
2nd
ⴙ
We reject the
negative root.
DISPLAY:
A(25.30, 28.48)
√(5.302+8.482)
10.00002
ENTER
Write the measure of OA to the nearest hundredth: OA 5 10.00.
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Similar Triangles
b. Let u be the measure of the angle in standard position with terminal side
h
OA. A is in the third quadrant. We can use either coordinate to find the
measure of u, a third-quadrant angle.
(OA cos u, OA sin u) 5 (25.30, 28.48)
10.00 sin u 5 28.48
10.00 cos u 5 25.30
sin u 5 20.848
cos u 5 20.530
ENTER:
2nd
)
DISPLAY:
COSⴚ1
(-)
0.530
ENTER:
2nd
)
ENTER
cos-1(-0.530)
122.0054548
The calculator returns the measure
of a second-quadrant angle. Use this
measure to find the reference angle
to the nearest degree.
DISPLAY:
SINⴚ1
(-)
0.848
ENTER
sin-1(-.848)
-57.99480014
The calculator returns the measure
of a fourth-quadrant angle. Use this
measure to find the reference angle
to the nearest degree.
R 5 2(258°) 5 58°
R 5 180 2 122 5 58°
Use the reference angle to find the measure of the third-quadrant angle.
u 5 180 1 58 5 238°
Answers a. OA 5 10.00
b. u 5 238°
Exercises
Writing About Mathematics
1. In Example 2, is it possible to find the measure of u without first finding OA? Justify your
answer.
4p
2. In what quadrant is a point whose coordinates in radian measure are A 2 cos 4p
3 , 2 sin 3 B ?
Justify your answer.
Developing Skills
In 3–14, write in simplest radical form the coordinates of each point A if A is on the terminal side
of an angle in standard position whose degree measure is u.
3. OA 5 4, u 5 45°
4. OA 5 2, u 5 30°
5. OA 5 6, u 5 90°
6. OA 5 8, u 5 120°
7. OA 5 15, u 5 135°
8. OA 5 0.5, u 5 180°
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Trigonometric Applications
9. OA 5 9, u 5 150°
10. OA 5 25, u 5 210°
12. OA 5 !2, u 5 225°
11. OA 5 12, u 5 270°
13. OA 5 !3, u 5 300°
14. OA 5 2, u 5 –60°
In 15–23, the coordinates of a point are given. a. Find the distance of the point from the origin.
Express approximate distances to the nearest hundredth. b. Find the measure, to the nearest degree,
of the angle in standard position whose terminal side contains the given point.
15. (6, 8)
16. (25, 12)
17. (0, 7)
18. (12, 29)
19. (15, 0)
20. (28, 212)
21. (24, 7)
22. (6, 210)
23. (28, 8)
In 24–29, for each ORS, O is the origin, R is on the positive ray of the x-axis and PS is the altitude
g
from S to OR. a. Find the exact coordinates of R and S. b. Find the exact area of ORS.
24. OR 5 5, mROS 5 p3 , OS 5 3
25. OR 5 12, mROS 5 p2 , OS 5 8
26. OR 5 8, mROS 5 3p
4 , OS 5 8
27. OR 5 20, OS 5 RS, PS 5 10
28. OR 5 9, ORS is equilateral
29. OR 5 7, mROS 5 p6 , PS 5 8
14-2 LAW OF COSINES
When we know the measures of two sides and the included angle of a triangle
(SAS), the size and shape of the triangle are determined. Therefore, we should
be able to find the measure of the third side of the triangle. In order to derive a
formula to do this, we will position the triangle in the coordinate plane with one
endpoint at the origin and one angle in standard position. As shown in the diagrams, the angle in standard position can be either acute, obtuse, or right.
y
C(b cos A, b sin A)
b
O
a
a
c
A(0, 0) B(c, 0)
y
C(b cos A, b sin A)
b
b
x
O
c
A(0, 0) B(c, 0)
x
O
C(b cos A, b sin A)
5 (0, b sin A)
a
x
c
A(0, 0) B(c, 0)
Let ABC be a triangle with AB 5 c, BC 5 a, and CA 5 b. The coordinates
of A are (0, 0), of B are (c, 0), and of C are (b cos A, b sin A). If b, c, and mA
are known measures, then the coordinates of each vertex are known. We can
find a, the length of the third side of the triangle, by using the distance formula.
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Law of Cosines
553
The distance between the two points P(x1, y1) and Q(x2, y2) is given by the
formula:
PQ2 5 (x2 – x1)2 1 (y2 2 y1)2
Let P(x1, y1) 5 B(c, 0) and Q(x2, y2) 5 C(b cos A, b sin A).
BC2 5 (b cos A – c)2 1 (b sin A 2 0)2
5 b2 cos2 A 2 2bc cos A 1 c2 1 b2 sin2 A
5 b2 cos2 A 1 b2 sin2 A 1 c2 2 2bc cos A
5 b2 (cos2 A 1 sin2 A) 1 c2 2 2bc cos A
5 b2(1) 1 c2 2 2bc cos A
5 b2 1 c2 2 2bc cos A
If we let BC 5 a, we can write:
a2 5 b2 1 c2 2 2bc cos A
This formula is called the Law of Cosines. The law of cosines for ABC
can be written in terms of the measures of any two sides and the included
angle.
a2 5 b2 1 c2 2 2bc cos A
b2 5 a2 1 c2 2 2ac cos B
c2 5 a2 1 b2 2 2ab cos C
We can rewrite the Law of Cosines in terms of
the letters that represent the vertices of any triangle. For example, in DEF, side DE is opposite
F so we let DE 5 f, side EF is opposite D so
we let EF 5 d, and side DF is opposite E so we
let DF 5 e. We can use the Law of Cosines to
write a formula for the square of the measure of
each side of DEF terms of the measures of the
other two sides and the included angle.
d 2 5 e 2 1 f 2 2 2ef cos D
e2 5 d2 1 f 2 2 2df cos E
f 2 5 d2 1 e2 2 2de cos F
F
d
e
D
f
E
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Trigonometric Applications
EXAMPLE 1
In ABC, AB 5 8, AC 5 10, and cos A 5 81. Find BC.
Solution AB 5 c 5 8, AC 5 b 5 10, and cos A 5 18.
A
Use the Law of Cosines to find BC 5 a.
10
a2 5 b2 1 c2 2 2bc cos A
a2 5 102 1 82 2 2(10)(8) A 18 B
8
a2 5 100 1 64 2 20
a
C
a2 5 144
B
a 5 612
Since a is the length of a line segment, a is a positive number.
Answer BC 5 12
EXAMPLE 2
The diagonals of a parallelogram measure 12 centimeters and 22 centimeters
and intersect at an angle of 143 degrees. Find the length of the longer sides of
the parallelogram to the nearest tenth of a centimeter.
Solution Let the diagonals of parallelogram PQRS inter- S
sect at T. The diagonals of a parallelogram bisect
each other. If PR 5 12, then PT 5 6 and if
QS 5 22, then QT 5 11. Let PQ be the longer
side of the parallelogram, the side opposite the
P
larger angle at which the diagonals intersect.
Therefore, mPTQ 5 143. Write the Law of
Cosines for PQ2 5 t 2 in terms of QT 5 p 5 11,
PT 5 q 5 6, and cos T 5 cos 143°.
11
t 2 5 p2 1 q2 2 2pq cos T
5 112 1 62 2 2(11)(6) cos 143°
5 121 1 36 2 132 cos 143°
t 5 !157 2 132 cos 1438
Note that cos 143° is negative so 2132 cos 143° is positive.
Using a calculator, we find that t 16.19937923.
Answer To the nearest tenth, PQ 5 16.2 cm.
T
6
143°
t
6
R
11
Q
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Law of Cosines
555
Exercises
Writing About Mathematics
1. Explain how the Law of Cosines can be used to show that in an obtuse triangle, the side
opposite an obtuse angle is the longest side of the triangle.
2. Explain the relationship between the Law of Cosines and the Pythagorean Theorem.
Developing Skills
3. In MAR, express m2 in terms of a, r, and cos M.
4. In NOP, express p2 in terms of n, o, and cos P.
5. In ABC, if a 5 3, b 5 5, and cos C 5 15, find the exact value of c.
6. In DEF, if e 5 8, f 5 3, and cos D 5 34, find the exact value of d.
7. In HIJ, if h 5 10, j 5 7, and cos I 5 0.6, find the exact value of i.
In 8–13, find the exact value of the third side of each triangle.
8. In ABC, b 5 4, c 5 4, and mA 5 p3 .
9. In PQR, p 5 6, q 5 !2, and mR 5 p4 .
10. In DEF, d 5 !3, e 5 5, and mF 5 p6 .
11. In ABC, a 5 6, b 5 4, and mC 5 2p
3.
12. In RST, RS 5 9, ST 5 9 !3, and mS 5 5p
6.
13. In ABC, AB 5 2 !2, BC 5 4, and mB 5 3p
4.
In 14–19, find, to the nearest tenth, the measure of the third side of each triangle.
14. In ABC, b 5 12.4, c 5 8.70, and mA 5 23.
15. In PQR, p 5 126, q 5 214, and mR 5 42.
16. In DEF, d 5 3.25, e 5 5.62, and mF 5 58.
17. In ABC, a 5 62.5, b 5 44.7, and mC 5 133.
18. In RST, RS 5 0.375, ST 5 1.29, and mS 5 167.
19. In ABC, AB 5 2.35, BC 5 6.24, and mB 5 115.
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Applying Skills
20. Ann and Bill Bekebrede follow a familiar triangular path when they take a walk. They walk
from home for 0.52 mile along a straight road, turn at an angle of 95°, walk for another 0.46
mile, and then return home.
a. Find, to the nearest hundredth of a mile, the length of the last portion of their walk.
b. Find, to the nearest hundredth of a mile, the total distance that they walk.
21. When two forces act on an object, the resultant force is the single force that would have produced the same result. When the
magnitudes of the two forces are represented by the lengths of
two sides of a parallelogram, the resultant can be represented
by the length of the diagonal of the parallelogram. If forces of
12 pounds and 18 pounds act at an angle of 75°, what is the
magnitude of the resultant force to the nearest hundredth pound?
12
75°
18
22. A field is in the shape of a parallelogram. The lengths of two adjacent sides are 48 meters
and 65 meters. The measure of one angle of the parallelogram is 100°.
a. Find, to the nearest meter, the length of the longer diagonal.
b. Find, to the nearest meter, the length of the shorter diagonal.
23. A pole is braced by two wires that extend from the top of the pole to the ground. The
lengths of the wires are 16 feet and 18 feet and the measure of the angle between the wires
is 110°. Find, to the nearest foot, the distance between the points at which the wires are fastened to the ground.
24. Two points A and B are on the shoreline of Lake George. A surveyor is located at a third
point C some distance from both points. The distance from A to C is 180.0 meters and the
distance from B to C is 120.0 meters. The surveyor determines that the measure of ACB is
56.3°. To the nearest tenth of a meter, what is the distance from A to B?
25. Two sailboats leave a dock at the same time sailing on courses that form an angle
of 112° with each other. If one boat sails at 10.0 knots and the other sails at
12.0 knots, how many nautical miles apart are the boats after two hours?
(nautical miles 5 knots 3 time) Round to the nearest tenth.
112°
26. Use the Law of Cosines to prove that if the angle between two congruent sides of a triangle
measures 60°, the triangle is equilateral.
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Using the Law of Cosines to Find Angle Measure
557
14-3 USING THE LAW OF COSINES TO FIND ANGLE MEASURE
The measures of three sides of a triangle determine the size and shape of the triangle. If we know the measures of three sides of a triangle, we can use the Law
of Cosines to find the measure of any angle of the triangle. For example, in
ABC, if a 5 7, b 5 5, and c 5 8, use the Law of Cosines to find cos A.
a2 5 b2 1 c2 2 2bc cos A
B
72 5 55 1 82 2 2(5)(8) cos A
7
49 5 25 1 64 2 80 cos A
C
8
5
80 cos A 5 25 1 64 2 49
80 cos A 5 40
cos A 5 40
80
A
cos A 5 12
Since A is an angle of a triangle, 0° , A , 180°. Therefore, A 5 60°.
The steps used to solve for cos A in terms of the measures of the sides can
be applied to the general formula of the Law of Cosines to express the cosine
of any angle of the triangle in terms of the lengths of the sides.
a2 5 b2 1 c2 2 2bc cos A
2bc cos A 5 b2 1 c2 2 a2
2
c2 2 a2
cos A 5 b 1 2bc
This formula can be rewritten in terms of the cosine of any angle of ABC.
2
c 2 2 a2
cos A 5 b 1 2bc
2
c 2 2 b2
cos B 5 a 12ac
2
b2 2 c 2
cos C 5 a 12ab
EXAMPLE 1
In ABC, a 5 12, b 5 8, c 5 6. Find cos C.
How to Proceed
Solution
(1) Write the Law of Cosines in terms of cos C:
(2) Substitute the given values:
(3) Perform the computation. Reduce the fractional
value of cos C to lowest terms:
Answer cos C 5
43
48
2
b2 2 c2
cos C 5 a 1 2ab
2 1 82 2 62
cos C 5 12 2(12)(8)
64 2 36
cos C 5 144 1192
5 172
192
5 43
48
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EXAMPLE 2
Find, to the nearest degree, the measure of the largest angle of DEF if
DE 5 7.5, EF 5 9.6, and DF 5 13.5.
Solution The largest angle of the triangle is opposite
the longest side. The largest angle is E, the
angle opposite the longest side, DF. Let
DE 5 f 5 7.5, EF 5 d 5 9.6, and
DF 5 e 5 13.5. Write the formula in terms
of cos E.
cos E 5
5
D
13.5
7.5
d2 1 f2 2 e2
2df
9.62 1 7.52 2 13.52
2(9.6)(7.5)
E
9.6
< 20.235
Therefore, E 5 arccos (20.235). Use a calculator to find the arccosine:
ENTER:
2nd
)
COSⴚ1
(-)
DISPLAY:
0.235
ENTER
cos-1(-0.235)
103.5916228
Answer mE 5 104
Exercises
Writing About Mathematics
1. Explain how the Law of Cosines can be used to show that 4, 7, and 12 cannot be the measures of the sides of a triangle.
2. Show that if C is an obtuse angle, a2 1 b2 , c2.
Developing Skills
3. In TUV, express cos T in terms of t, u, and v.
4. In PQR, express cos Q in terms of p, q, and r.
5. In KLM, if k 5 4, l 5 5, and m 5 8, find the exact value of cos M.
6. In XYZ, if x 5 1, y 5 2, and z 5 !5, find the exact value of cos Z.
In 7–12, find the cosine of each angle of the given triangle.
7. In ABC, a 5 4, b 5 6, c 5 8.
9. In DEF, d 5 15, e 5 12, f 5 8.
11. In MNP, m 5 16, n 5 15, p 5 8.
8. In ABC, a 5 12, b 5 8, c 5 8.
10. In PQR, p 5 2, q 5 4, r 5 5.
12. In ABC, a 5 5, b 5 12, c 5 13.
F
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Area of a Triangle
In 13–18, find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides.
13. 12, 20, 22
14. 9, 10, 15
15. 30, 35, 45
16. 11, 11, 15
17. 32, 40, 38
18. 7, 24, 25
Applying Skills
19. Two lighthouses are 12 miles apart along a straight shore. A ship is 15 miles from one lighthouse and 20 miles from the other. Find, to the nearest degree, the measure of the angle
between the lines of sight from the ship to each lighthouse.
20. A tree is braced by wires 4.2 feet and 4.7 feet long that are fastened to the tree at the same
point and to the ground at points 7.8 feet apart. Find, to the nearest degree, the measure of
the angle between the wires at the tree.
21. A kite is in the shape of a quadrilateral with two pair of congruent adjacent sides. The
lengths of two sides are 20.0 inches and the lengths of the other two sides are 35.0 inches.
The two shorter sides meet at an angle of 115°.
a. Find the length of the diagonal between the points at which the unequal sides meet.
Write the length to the nearest tenth of an inch.
b. Using the answer to part a, find, to the nearest degree, the measure of the angle at
which the two longer sides meet.
22. A beam 16.5 feet long supports a roof with rafters each measuring
12.4 feet long. What is the measure of the angle at which the rafters
meet?
4
2.
ft
x
1
12
.4
ft
16.5 ft
23. A walking trail is laid out in the shape of a triangle. The lengths of the three paths that
make up the trail are 2,500 meters, 2,000 meters, and 1,800 meters. Determine, to the nearest
degree, the measure of the greatest angle of the trail.
2
b2 2 c2
24. Use the formula cos C 5 a 1 2ab
to show that the measure of each angle of an equilateral triangle is 60°.
14-4 AREA OF A TRIANGLE
When the measures of two sides and the included angle of a triangle are known,
the size and shape of the triangle is determined. Therefore, it is possible to use
these known values to find the area of the triangle. Let ABC be any triangle.
If we know the measures of AB 5 c, AC 5 b, and the included angle, A, we
can find the area of the triangle.
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Trigonometric Applications
B
B
c
c
h
h
D
A
C
A
b
C
D
b
g
In ABC let A be an acute angle, BD be the altitude from B to AC , and
opp
h
BD 5 h. In right ABD, sin A 5 hyp 5 BD
AB 5 c or h 5 c sin A. Therefore:
Area of ABC 5 12bh 5 12bc sin A
What if A is an obtuse angle?
B
Let A be an obtuse angle of ABC, BD be the
g
altitude from B to AC and BD 5 h. In right ABD, h
opp
h
let mDAB 5 u. Then sin u 5 hyp 5 BD
AB 5 c or
h 5 c sin u. Therefore:
D
c
u
A
b
C
Area of ABC 5 12bh 5 12bc sin u
Since DAB and BAC are adjacent angles whose sum is a straight angle,
mDAB 5 180 2 mBAC and sin u 5 sin A. Therefore, the area of ABC is
again equal to 12bc sin A. Thus, for any angle, we have shown that:
Area of ABC 5 12bh 5 12bc sin A
The area of a triangle is equal to one-half the product of the measures of
two sides of the triangle times the sine of the measure of the included angle. This
formula can be written in terms of any two sides and the included angle.
Area nABC 5 12bc sin A 5 12ac sin B 5 12ab sin C
Triangles in the Coordinate Plane
When a triangle is drawn in the coordinate plane, the area formula follows
easily. Let ABC be any triangle. Place the triangle with A(0, 0) at the origin
and C(b, 0) on the positive ray of the x-axis, and A an angle in standard
position.
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Area of a Triangle
y
y
y
B(c cos A, c sin A)
B(c cos A, c sin A)
O
B(c cos A, c sin A)
h
c
h
A(0, 0) C(b, 0)
561
h
x
x
O
A(0, 0)
C(b, 0)
O
x
A(0, 0) C(b, 0)
From Section 14-1, we know that the coordinates of B are (c cos A, c sin A).
For each triangle, h is the length of the perpendicular from B to the x-axis and
h 5 c sin A. Therefore,
Area ABC 5 12bh 5 12bc sin A
EXAMPLE 1
Find the area of DEF if DE 5 14, EF 5 9, and mE 5 30.
Solution DE 5 f 5 14, EF 5 d 5 9, and mE 5 30.
1
Area of DEF 5 12df sin E 5 21 (9)(14) A 12 B 5 63
2 5 312 Answer
EXAMPLE 2
The adjacent sides of parallelogram ABCD measure 12 and 15. The measure of
one angle of the parallelogram is 135°. Find the area of the parallelogram.
Solution The diagonal of a parallelogram separates
the parallelogram into two congruent triangles. Draw diagonal BD. In DAB,
DA 5 b 5 12, AB 5 d 5 15, and
mA 5 135.
C
D
12
135°
A
15
Area of DAB 5 12bd sin A 5 12 (12)(15) sin 1358 5 12 (12)(15) Q !2
2 R 5 45 !2
Area of DBC 5 Area of DAB 5 45 !2
Area of parallelogram ABCD 5 Area of DBC 1 Area of DAB 5 90!2
Answer 90 !2 square units
B
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Note: The same answer to Example 2 is
obtained if we use adjacent angle B or D.
Consecutive angles of a parallelogram are
supplementary. If mA 5 135, then
mB 5 180 2 135 5 45. Opposite sides of a
parallelogram are congruent. If DA 5 12,
then BC 5 12. Draw diagonal AC. In ABC,
BC 5 a 5 12, AB 5 c 5 15, and mB 5 45.
C
D
12
45°
A
B
15
Area of ABC 5 12ac sin B 5 12 (12)(15) sin 458 5 12 (12)(15) Q !2
2 R 5 45!2
Area of CDA 5 Area of ABC 5 45 !2
Area of parallelogram ABCD 5 Area of ABC 1 Area of CDA 5 90!2
EXAMPLE 3
Three streets intersect in pairs enclosing a small triangular park. The measures
of the distances between the intersections are 85.5 feet, 102 feet, and 78.2 feet.
Find the area of the park to the nearest ten square feet.
Solution Let A, B, and C be the intersections of
the streets, forming ABC. Use the
Law of Cosines to find the measure of
any angle, for example, A. Then use
the formula for the area of a triangle in
terms of the measures of two sides and
an angle.
C
78.2 ft
85.5 ft
B
A
102 ft
2
2 1 1022 2 78.22
c2 2 a2
cos A 5 b 1 2bc
5 85.5 2(85.5)(102)
0.6650
Use a calculator to find the measure of A.
mA 5 cos–1 0.6650 48.31
Area of ABC 5 21bc sin A 5 12(85.5)(102)(sin 48.31°) 3,256
Answer The area of the park is approximately 3,260 square feet.
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Area of a Triangle
563
Exercises
Writing About Mathematics
1. Rosa found the area of parallelogram ABCD by using (AB)(BC)(sin B). Riley found the
area of parallelogram ABCD by using (AB)(BC)(sin A). Explain why Rosa and Riley both
got the correct answer.
2. Jessica said that the area of rhombus PQRS is (PQ)2(sin P). Do you agree with Jessica?
Explain why or why not.
Developing Skills
In 3–8, find the area of each ABC.
3. b 5 3, c 5 8, sin A 5 41
4. a 5 12, c 5 15, sin B 5 13
5. b 5 9, c 5 16, sin A 5 65
6. a 5 24, b 5 12, sin C 5 43
7. b 5 7, c 5 8, sin A 5 53
3
8. a 5 10, c 5 8, sin B 5 10
In 9–14, find the area of each triangle to the nearest tenth.
9. In ABC, b 5 14.6, c 5 12.8, mA 5 56.
10. In ABC, a 5 326, c 5 157, mB 5 72.
11. In DEF, d 5 5.83, e 5 5.83, mF 5 48.
12. In PQR, p 5 212, q 5 287, mR 5 124.
13. In RST, t 5 15.7, s 5 15.7, mR 5 98.
14. In DEF, e 5 336, f 5 257, mD 5 122.
15. Find the exact value of the area of an equilateral triangle if the length of one side is 40
meters.
16. Find the exact value of the area of an isosceles triangle if the measure of a leg is 12 centimeters and the measure of the vertex angle is 45 degrees.
17. Find the area of a parallelogram if the measures of two adjacent sides are 40 feet and 24
feet and the measure of one angle of the parallelogram is 30 degrees.
Applying Skills
18. A field is bordered by two pairs of parallel roads so that the shape of the field is a parallelogram. The lengths of two adjacent sides of the field are 2 kilometers and 3 kilometers, and
the length of the shorter diagonal of the field is 3 kilometers.
a. Find the cosine of the acute angle of the parallelogram.
b. Find the exact value of the sine of the acute angle of the parallelogram.
c. Find the exact value of the area of the field.
d. Find the area of the field to the nearest integer.
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19. The roof of a shed consists of four congruent isosceles triangles. The length of each equal
side of one triangular section is 22.0 feet and the measure of the vertex angle of each triangle is 75°. Find, to the nearest square foot, the area of one triangular section of the roof.
20. A garden is in the shape of an isosceles trapezoid. The lengths of the parallel sides of the
garden are 30 feet and 20 feet, and the length of each of the other two sides is 10 feet. If a
base angle of the trapezoid measures 60°, find the exact area of the garden.
21. In ABC, mB 5 30 and in DEF, mE 5 150. Show that if AB 5 DE and BC 5 EF, the
areas of the two triangles are equal.
22. Aaron wants to draw ABC with AB 5 15 inches, BC 5 8 inches, and an area of 40 square
inches.
a. What must be the sine of B?
b. Find, to the nearest tenth of a degree, the measure of B.
c. Is it possible for Aaron to draw two triangles that are not congruent to each other that
satisfy the given conditions? Explain.
D
23. Let ABCD be a parallelogram with AB 5 c, BC 5 a, and
mB 5 u.
C
a. Write a formula for the area of parallelogram ABCD in
terms of c, a, and u.
b. For what value of u does parallelogram ABCD have the
greatest area?
a
u
c
A
B
14-5 LAW OF SINES
If we know the measures of two angles and the included side of a triangle
(ASA), or if we know the measures of two angles and the side opposite one
of the angles of a triangle (AAS), the size and shape of the triangle is determined. Therefore, we should be able to find the measures of the remaining
sides.
In ABC, let mA and mB be two angles
C
and AC 5 b be the side opposite one of the angles.
When we know these measures, is it possible to find
BC 5 a?
a
b
h
A
D
B
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Law of Sines
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Let CD be the altitude from C to AB. Let CD 5 h and BC 5 a.
In right ACD,
sin A 5
In right BCD,
opp
opp
hyp
sin B 5 hyp
sin A 5 hb
sin B 5 ha
h 5 b sin A
h 5 a sin B
Since a sin B and b sin A are each equal to h, they are equal to each other.
Therefore, a sin B 5 b sin A. To solve for a, divide both sides of this equation by
sin B.
a sin B 5 b sin A
a sin B
sin B
sin A
5 bsin
B
sin A
a 5 bsin
B
More generally, we can establish a proportional relationship between two
angles and the sides opposite these angles in a triangle. Divide both sides of this
equation by sin A sin B.
a sin B 5 b sin A
a sin B
sin A sin B
a
sin A
sin A
5 sinb A
sin B
5 sinb B
An alternative derivation of this formula begins with the formulas for the
area of a triangle.
Area ABC 5 12bc sin A 5 12ac sin B 5 12ab sin C
We can multiply each of the last three terms of this equality by 2.
2 A 12bc sin A B 5 2 A 12ac sin B B 5 2 A 12ab sin C B
bc sin A 5 ac sin B 5 ab sin C
Now divide each side of the equality by abc.
bc sin A
abc
sin C
sin B
5 acabc
5 ababc
sin A
a
5 sinb B 5 sinc C
These equal ratios are usually written in terms of their reciprocals.
a
sin A
5 sinb B 5 sinc C
This equality is called the Law of Sines.
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Trigonometric Applications
EXAMPLE 1
In ABC, c 5 12, mB 5 120, and mC 5 45. Find
the exact value of b.
Solution
B
C
12
How to Proceed
b
sin B
(1) Use the ratios of the
Law of Sines that use
b and c:
5 sinc C
A
b
sin 1208
(2) Substitute the given
values:
5
12
sin 458
sin 1208
b 5 12sin
458
(3) Solve for b, substituting
sine values:
b5
(4) Write the value of b in
simplest form:
b5
12Q !3
2 R
!2
2
12 !3
2
2
3 !2
!3
5 12!2
3 !2
!2
5 122!6
5 6 !6
Answer b 5 6 !6
EXAMPLE 2
In DEF, mD 5 50, mE 5 95, and f 5 12.6.
Find d to the nearest tenth.
F
E
95°
Solution Use the form of the Law of Sines in terms of 12.6
the side whose measure is known, f, and the
side whose measure is to be found, d.
50°
d
sin D
f
5 sin F
D
To use this formula, we need to know mF.
mF 5 180 2 (50 1 95) 5 35
Therefore:
d
sin 508
12.6
5 sin
358
sin 508
d 5 12.6
sin 358
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Law of Sines
567
Use a calculator to evaluate d.
ENTER:
(
ⴜ
12.6 SIN
SIN
35
50
)
)
ENTER
)
DISPLAY:
(12.6 sin(50))/si
n(35)
16.82802739
Answer To the nearest tenth, d 5 16.8.
Exercises
Writing About Mathematics
1. If the sine of an angle of a triangle is known, is it possible to determine the measure of the
angle? Explain why or why not.
2. If the cosine of an angle of a triangle is known, is it possible to determine the measure of
the angle? Explain why or why not.
Developing Skills
3. In ABC, if a 5 9, mA 5 p3 , and mB 5 p4 , find the exact value of b in simplest form.
4. In ABC, if a 5 24, mA 5 p6 , and mB 5 p2 , find the exact value of b in simplest form.
p
5. In ABC, if c 5 12, mC 5 2p
3 , and mB 5 6 , find the exact value of b in simplest form.
6. In ABC, if b 5 8, mA 5 p3 , and mC 5 5p
12 , find the exact value of a in simplest form.
7. In DEF, sin D 5 0.4, sin E 5 0.25, and d 5 20. Find the exact value of e.
8. In PQR, sin P 5 43, sin R 5 25, and p 5 40. Find the exact value of r.
9. In DEF, mD 5 47, mE 5 84, and d 5 17.3. Find e to the nearest tenth.
10. In DEF, mD 5 56, mE 5 44, and d 5 37.5. Find e to the nearest tenth.
11. In LMN, mM 5 112, mN 5 54, and m 5 51.0. Find n to the nearest tenth.
12. In ABC, mA 5 102, mB 5 34, and a 5 25.8. Find c to the nearest tenth.
13. In PQR, mP 5 125, mQ 5 14, and p 5 122. Find r to the nearest integer.
14. In RST, mR 5 12, mS 5 75, and r 5 3.52. Find t to the nearest tenth.
15. In CDE, mD 5 125, mE 5 28, and d 5 12.5. Find c to the nearest hundredth.
16. The base of an isosceles triangle measures 14.5 centimeters and the vertex angle measures
110 degrees.
a. Find the measure of one of the congruent sides of the triangle to the nearest hundredth.
b. Find the perimeter of the triangle to the nearest tenth.
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Trigonometric Applications
17. The length of one of the equal sides of an isosceles triangle measures 25.8 inches and each
base angle measures 53 degrees.
a. Find the measure of the base of the triangle to the nearest tenth.
b. Find the perimeter of the triangle to the nearest inch.
18. Use the Law of Sines to show that if C of ABC is a right angle, sin A 5 ac.
Applying Skills
19. A telephone pole on a hillside makes an angle
of 78 degrees with the upward slope. A wire
from the top of the pole to a point up the hill is
12.0 feet long and makes an angle of 15 degrees
with the pole.
a. Find, to the nearest hundredth, the distance from
the foot of the pole to the point at which the wire
is fastened to the ground.
78°
b. Use the answer to part a to find, to the nearest
tenth, the height of the pole.
20. Three streets intersect in pairs enclosing a small park. Two of the angles at which the streets
intersect measure 85 degrees and 65 degrees. The length of the longest side of the park is
275 feet. Find the lengths of the other two sides of the park to the nearest tenth.
21. On the playground, the 10-foot ladder to the top of the slide makes an angle of 48 degrees
with the ground. The slide makes an angle of 32 degrees with the ground.
a. How long is the slide to the nearest tenth?
b. What is the distance from the foot of the ladder to the foot of the slide to the nearest
tenth?
22. A distress signal from a ship, S, is received by two coast guard
stations located 3.8 miles apart along a straight coastline. From
station A, the signal makes an angle of 48° with the coastline
and from station B the signal makes an angle of 67° with
the coastline. Find, to the nearest tenth of a mile, the
distance from the ship to the nearer station.
A
48°
S
67°
B
23. Two sides of a triangular lot form angles that measure 29.1° and 33.7° with the third side,
which is 487 feet long. To the nearest dollar, how much will it cost to fence the lot if the
fencing costs $5.59 per foot?
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The Ambiguous Case
14-6 THE AMBIGUOUS CASE
If we know the measures of two sides of a triangle and the angle opposite one
of them, the Law of Sines makes it possible for us to find the sine of the angle
opposite the second side whose measure is known. However, we know that the
measures of two sides and the angle opposite one of them (SSA) is not sufficient
to determine the size and shape of the triangle in every case. This is often called
the ambiguous case.
Consider the following cases in which we are given a, b, and angle A. For
0 , sin B , 1, there are two values of B in the interval from 0° to 180°. We will
call these values B and B9. Since the sum of the degree measures of the angles
of a triangle is 180, the sum of the degree measures of two angles of a triangle
must be less than 180.
CASE 1
Two triangles can be drawn.
In ABC, a 5 8, b 5 12, and A 5 30°.
We can use the Law of Sines to find sin B.
a
sin A
8
sin 308
B
5 sinb B
12
5 sin
B
8
8 sin B 5 12 sin 30°
sin B 5
12 A 12 B
8
8
B9
30°
C
12
A
sin B 5 34
When sin B 5 34, mB 48.59 or mB9 180 2 48.59 5 131.41. As shown
in the diagram, there are two triangles, ABC and AB9C in which two sides
measure 8 and 12 and the angle opposite the shorter of these sides measures
30°. Two triangles can be drawn.
CASE 2
Only one triangle can be drawn and that triangle is a right triangle.
In ABC, a 5 8, b 5 16, and A 5 30°.
We can use the Law of Sines to find sin B.
a
sin A
8
sin 308
C
5 sinb B
16
16
5 sin
B
8
8 sin B 5 16 sin 30°
sin B 5
16 A 12 B
8
sin B 5 1
30°
A
B
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Trigonometric Applications
When sin B 5 1, mB 5 90. This is the only measure of B that can be the
measure of an angle of a triangle. One triangle can be drawn and that triangle
is a right triangle.
Note: If mA 5 150, sin B 5 1 and mB 5 90. There is no triangle with an
obtuse angle and a right angle.
CASE 3
Only one triangle can drawn.
In ABC, a 5 16, b 5 8, and A 5 30°. We can use the Law of Sines to
find sin B.
a
sin A
16
sin 308
5 sinb B
5 sin8 B
16 sin B 5 8 sin 30°
8A1B
sin B 5 162
C
8
16
30°
A
B
sin B 5 41
When sin B 5 14, mB 5 14.48 or mB 5 180 2 14.48 5 165.52.
• If we let B be an acute angle, mA 1 m B 5 30 1 14.48 , 180. There
is a triangle with mA 5 30 and mB 5 14.48. ✔
• If we let B be an obtuse angle, mA 1 mB 5 30 1 165.52 . 180.
There is no triangle with mA 5 30 and mB 5 165.52. ✘
Only one triangle can be drawn.
CASE 4
No triangle can be drawn.
In ABC, a 5 8, b 5 20, and A 5 30°. We can use the Law of Sines to
find sin B.
C
b
a
sin A 5 sin B
8
sin 308
20
5 sin
B
8 sin B 5 20 sin 30°
sin B 5
20 A 12 B
8
sin B 5 54
8
20
B
30°
A
There is no value of B for which sin B . 1. No triangle can be drawn.
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The Ambiguous Case
571
These four examples show that for the given lengths of two sides and the
measure of an acute angle opposite one of them, two, one, or no triangles can be
formed.
To determine the number of solutions given a, b,
and m * A in ABC:
Use the Law of Sines to solve for sin B.
If sin B + 1, there is no triangle.
If sin B 5 1, there is one right triangle
if * A is acute but no triangle if * A
is obtuse.
C
C
C
B
B
A
A
A
B
If * A is acute and sin B * 1, find two possible values of * B:
0 * m * B , 90 and m * B9 5 180 2 m * B.
C
C
B9
A
B
If mA 1 mB9 , 180, there
are two possible triangles,
ABC and AB9C.
B
A
If mA 1 mB9 $ 180, AB9C is
not a triangle. There is only one
possible triangle, ABC.
If * A is obtuse and sin B * 1, * B must be acute: 0 * m * B * 90.
C
C
B
A
B
If mA 1 mB , 180, there is
one triangle, ABC.
A
If mA 1 mB $ 180, there is no
triangle.
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Trigonometric Applications
Alternatively, if we let h 5 b sin A, the height of the triangle, we can summarize the number of possible triangles given a, b, and mA in ABC:
C
C
C
C
b
h
Possible
triangles:
Acute
a,h
None
b
ha
A
A
A is:
b
b
h
a h a
B
B
A
Acute
h5a
One,
right B9
C
C
a
B
a
h
A
Acute
h,a,b
Two
a
b
a
B
h b
A
A
B
Acute
a.b
One
Obtuse
a#b
None
B
Obtuse
a.b
One
EXAMPLE 1
In ABC, b 5 9, c 5 12, and mC 5 45.
B
a. Find the exact value of sin B.
b. For the value of sin B in a, find, to the nearest hundredth, the measures of two angles, B and B9,
that could be angles of a triangle.
c. How many triangles are possible?
12
C
Solution a. (1) Use the ratios of the Law of Sines that use
b and c:
b
sin B
45°
9
A
5 sinc C
9
sin B
5 sin12458
12 sin B 5 9 sin 458
(2) Substitute the given values:
(3) Solve for sin B:
9
sin B 5 12
sin 458
(4) Substitute the exact value of sin 45:
sin B 5 34 3 !2
2
sin B 5 3 !2
8
b. Use a calculator to find the approximate measure of B.
ENTER:
2nd
SINⴚ1
(
3 ⴛ
2nd
¯ 2
)
)
ⴜ
DISPLAY:
sin-1((3* √(2))/8
32.02776011
8 ENTER
mB 32.03 and mB9 180 2 32.03 5 147.97 Answer
c. mA 1 mB 5 45 1 32.03 , 180 and ABC is a triangle.
mA 1 mB9 5 45 1 147.97 . 180 and AB9C is not a triangle.
There is one possible triangle. Answer
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The Ambiguous Case
573
EXAMPLE 2
How many triangles can be drawn if the measures of two of the sides are 12
inches and 10 inches and the measure of the angle opposite the 10-inch side is
110 degrees?
Q
Solution Let a possible triangle be PQR with
PQ 5 r 5 12 and QR 5 p 5 10. The angle
opposite QR is P and mP 5 110.
10
12
R
110°
P
(1) We know p, r, and P. Use the
Law of Sines to solve for sin R:
p
sin P
10
sin 1108
5 sinr R
12
5 sin
R
10 sin R 5 12 sin 110°
1108
sin R 5 12 sin
10
(2) Use a calculator to find sin R:
sin R 1.127631145
Since sin R . 1, there is no triangle.
Alternative Let h represent the height of the triangle. Then
Solution
h 5 r sin P 5 12 sin 70° 11.28
Q
The height of the triangle, or the altitude at Q,
would be longer than side QR. No such triangle exists.
h
Answer 0
10
12
R
70° 110°
P
Exercises
Writing About Mathematics
1. Without using formulas that include the sine of an angle, is it possible to determine from the
given information in Example 2 that there can be no possible triangle? Justify your answer.
2. Explain why, when the measures of two sides and an obtuse angle opposite one of them are
given, it is never possible to construct two different triangles.
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Trigonometric Applications
Developing Skills
In 3–14: a. Determine the number of possible triangles for each set of given measures. b. Find the
measures of the three angles of each possible triangle. Express approximate values to the nearest
degree.
3. a 5 8, b 5 10, mA 5 20
4. a 5 5, b 5 10, mA 5 30
5. b 5 12, c 5 10, mB 5 49
6. a 5 6, c 5 10, mA 5 45
7. c 5 18, b 5 10, mC 5 120
8. a 5 9, c 5 10, mC 5 150
9. AB 5 14, BC 5 21, mC 5 75
10. DE 5 24, EF 5 18, mD 5 15
11. PQ 5 12, PR 5 15, mR 5 100
12. BC 5 12, AC 5 12 !2, mB 5 135
13. RS 5 3!3, ST 5 3, mT 5 60
14. a 5 8, b 5 10, mA 5 45
Applying Skills
15. A ladder that is 15 feet long is placed so that it reaches from level ground to the top of a
vertical wall that is 13 feet high.
a. Use the Law of Sines to find the angle that the ladder makes with the ground to the
nearest hundredth.
b. Is more than one position of the ladder possible? Explain your answer.
16. Max has a triangular garden. He measured two sides of the garden and the angle opposite
one of these sides. He said that the two sides measured 5 feet and 8 feet and that the angle
opposite the 8-foot side measured 75 degrees. Can a garden exist with these measurements?
Could there be two gardens of different shapes with these measurements? Write the angle
measures and lengths of the sides of the garden(s) if any.
17. Emily wants to draw a parallelogram with the measure of one side 12 centimeters, the measure of one diagonal 10 centimeters and the measure of one angle 120 degrees. Is this possible? Explain why or why not.
18. Ross said that when he jogs, his path forms a triangle. Two sides of the triangle are 2.0 kilometers and 2.5 kilometers in length and the angle opposite the shorter side measures 45
degrees. Rosa said that when she jogs, her path also forms a triangle with two sides of length
2.0 kilometers and 2.5 kilometers and an angle of 45 degrees opposite the shorter side. Rosa
said that her route is longer than the route Ross follows. Is this possible? Explain your
answer.
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Solving Triangles
575
14-7 SOLVING TRIANGLES
If the known measures of any three parts of a triangle include at least one side,
the measures of the remaining three parts can be determined.
The Right Triangle
When the triangle is a right triangle, the ratio of the measures of any two sides
of the triangle is a trigonometric function value of one of the acute angles.
In right ABC with mC 5 90:
B
opp
BC
sin A 5 hyp 5 AB
adj
AC
cos A 5 hyp 5 AB
opp
BC
tan A 5 adj 5 AC
C
A
The height of a building, a tree, or any similar object is measured as the
length of the perpendicular from the top of the object to the ground. The measurement of height often involves right triangles.
An angle of elevation is an angle such
B
that one ray is part of a horizontal line and
the other ray represents a line of sight raised
upward from the horizontal. To visualize
the angle of elevation of a building, think of
some point on the same horizontal line as the
base of the building. The angle of elevation is
the angle through which our line of sight
angle of
elevation
would rotate from the base of the building
to its top. In the diagram, A is the angle of
A
C
elevation.
An angle of depression is an angle such
B
D
that one ray is part of a horizontal line and
angle of
the other ray represents a line of sight moved
depression
downward from the horizontal. To visualize
the angle of depression of a building, think of
some point on the same horizontal line as the
top of the building. The angle of depression is
the angle through which our line of sight
would rotate from the top of the building to
A
its base. In the diagram, ABD is the angle of
C
depression.
When solving right triangles, we can use the ratio of sides given above or we
can use the Law of Sines or the Law of Cosines.
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Trigonometric Applications
EXAMPLE 1
From a point 12 feet from the foot of the tree, the
measure of the angle of elevation to the top of
the tree is 57°. Find the height of the tree to the
nearest tenth of a foot.
C
h
Solution The height of the tree is the length of the
perpendicular from the top of the tree to the
ground. Use the ratio of sides of a right triangle.
We know the length of the side adjacent to the
angle of elevation and we want to know the height
of the tree, the length of the side opposite the angle
of elevation.
tan u 5
tan 57° 5
57°
A
12 ft
B
opp
adj
h
12
h 5 12 tan 57°
h 5 18.478
Alternative We know the measures of two angles and the included side. Find the measure
Solution of the third angle and use the Law of Sines.
mB 5 180 2 (mA 1 mC)
mB 5 180 2 (57 1 90)
mB 5 33
a
sin A
5 sinb B
h
sin 578
5 sin12338
578
h 5 12sinsin338
h 5 18.478
Answer The tree is 18.5 feet tall.
The General Triangle
The Law of Cosines and the Law of Sines can be used to find the remaining
three measures of any triangle when we know the measure of a side and the
measures of any two other parts of the triangle.
CASE 1
Given: Two sides and the included angle
• Use the Law of Cosines to find the measure of the third side.
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Solving Triangles
577
• Use the Law of Sines or the Law of Cosines to find the measure of
another angle.
• Use the sum of the angles of a triangle to find the measure of the third
angle.
CASE 2
Given: Three sides
• Use the Law of Cosines to find the measure of an angle.
• Use the Law of Sines or the Law of Cosines to find the measure of
another angle.
• Use the sum of the angles of a triangle to find the measure of the third
angle.
CASE 3
Given: Two angles and a side
• Use the sum of the angles of a triangle to find the measure of the third
angle.
• Use the Law of Sines to find the remaining sides.
CASE 4
Given: Two sides and an angle opposite one of them
• Use the Law of Sines to find the possible measure(s) of another angle.
• Determine if there are two, one, or no possible triangles.
• If there is a triangle, use the sum of the angles of a triangle to find the
measure(s) of the third angle.
• Use the Law of Sines or the Law of Cosines to find the measure(s) of the
third side.
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Trigonometric Applications
EXAMPLE 2
The Parks Department is laying out a nature trail through the woods that is to
consist of three straight paths that form a triangle. The lengths of two paths measure 1.2 miles and 1.5 miles. What must be the measure of the angle between
these two sections of the path in order that the total length of the nature trail
will be 4.0 miles?
B
Solution (1) Draw a diagram. The trail forms a
triangle, ABC, with AB 5 c 5 1.2 and
BC 5 a 5 1.5:
1.2
1.5
A
b
C
(2) The perimeter of the triangle is 4.0.
Find CA 5 b:
P 5 AB 1 BC 1 CA
4.0 5 1.2 1 1.5 1 b
4.0 5 2.7 1 b
1.3 5 b
(3) Use the Law of Cosines to find the
measure of an angle when the measures
of three sides are known. Find the measure
of B:
2
c2 2 b2
cos B 5 a 1 2ac
2 1 1.22 2 1.32
cos B 5 1.5 2(1.5)(1.2)
cos B 5 0.5555 . . .
mB 5 56.25
Answer 56°
EXAMPLE 3
A man, standing on a cliff 85 feet high at the edge of the water, sees two ships.
He positions himself so that the ships are in a straight line with a point directly
below where he is standing. He estimates that the angle of depression of the
closer ship is 75 degrees and the angle of depression of the farther ship is 35
degrees. How far apart are the two ships?
C
Solution (1) Draw and label a diagram. Let A be
35°
75°
the closer ship, B the farther ship, C
the edge of the cliff where the man is
85 ft
40°
standing, and D the point directly
below C at sea level. The angle of
depression is the angle between a
horizontal ray and a ray downward to
35°
105° 75°
the ship. Determine the measure of
B
D
A
each angle in the diagram:
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Solving Triangles
opp
(2) Use right ACD to find AC, the measure of
a side of both ABC and ACD.
sin 75° 5 hyp
85
sin 75° 5 AC
AC 5 sin85758
AC 87.998
AB
sin /ACB
AB
sin 408
(3) In ABC, we now know the measures of
two angles and the side opposite one of
them. Use the Law of Sines to find AB, the
required distance.
5 sin AC
/ABC
87.998
5 sin
358
sin 408
AB 5 87.998
sin 358
AB 98.62
Answer The ships are 99 feet apart.
Exercises
Writing About Mathematics
1. Navira said that in Example 3, it would have been possible to solve the problem by using
BDC to find BC first. Do you agree with Navira? Explain why or why not.
2. Explain why an angle of depression is always congruent to an angle of elevation.
Developing Skills
In 3–10: a. State whether each triangle should be solved by using the Law of Sines or the Law of
Cosines. b. Solve each triangle, rounding answers to the nearest tenth. Include all possible solutions.
3.
4.
7
33°
5.
12
9
7
12
x
30°
6.
63°
5.8
10
7.4
x
7.
6.1
x
x
x
5
8.
6
x
13
9.
x
10.
x
94°
30°
8
17
29°
56°
32°
24
16.8
In 11–22, solve each triangle, that is, find the measures of the remaining three parts of the triangle
to the nearest integer or the nearest degree.
11. In ABC, a 5 15, b 5 18, and mC 5 60.
12. In ABC, a 5 10, b 5 12, and mB 5 30.
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Trigonometric Applications
13. In ABC, b 5 25, mB 5 45, and mC 5 60.
14. In ABC, a 5 8, mB 5 35, and mC 5 55.
15. In DEF, d 5 72, e 5 48, and mF 5 110.
16. In PQR, p 5 12, mQ 5 80, and mR 5 30.
17. In RST, r 5 38, s 5 28, and t 5 18.
18. In ABC, a 5 22, b 5 18, and mC 5 130.
19. In PQR, p 5 12, q 5 16, and r 5 20.
20. In DEF, d 5 36, e 5 72, and mD 5 30.
21. In RST, r 5 15, s 5 18, and mT 5 90.
B
22. In ABC, a 5 15, b 5 25, and c 5 12.
23. In the diagram, AD 5 25, CD 5 10, BD 5 BC,
and mD 5 75. Find AB to the nearest tenth.
A
75°
C 10 D
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25
Applying Skills
24. A small park is in the shape of an isosceles trapezoid. The length of the longer of the parallel sides is 3.2 kilometers and the length of an adjacent side is 2.4 kilometers. A path from
one corner of the park to an opposite corner is 3.6 kilometers long.
a. Find, to the nearest tenth, the measure of each angle between adjacent sides of the park.
b. Find, to the nearest tenth, the measure of each angle between the path and a side of the
park.
c. Find, to the nearest tenth, the length of the shorter of the parallel sides.
25. From a point on the ground 50 feet from the foot of a vertical monument, the measure of
the angle of elevation of the top of the monument is 65 degrees. What is the height of the
monument to the nearest foot?
26. A vertical telephone pole that is 15 feet high is braced by two wires from the top of the pole
to two points on the ground that are 5.0 feet apart on the same side of the pole and in a
straight line with the foot of the pole. The shorter wire makes an angle of 65 degrees with
the ground. Find the length of each wire to the nearest tenth.
27. From point C at the top of a cliff, two points, A and B, are sited on level ground. Points A and
B are on a straight line with D, a point directly below C. The angle of depression of the nearer
point, A, is 72 degrees and the angle of depression of the farther point, B, is 48 degrees. If the
points A and B are 20 feet apart, what is the height of the cliff to the nearest foot?
28. Mark is building a kite that is a quadrilateral with two pairs of congruent adjacent sides. One
diagonal divides the kite into two unequal isosceles triangles and measures 14 inches. Each
leg of one of the isosceles triangles measures 15 inches and each leg of the other measures
12 inches. Find the measures of the four angles of the quadrilateral to the nearest tenth.
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Chapter Summary
581
CHAPTER SUMMARY
When we know the measures of three sides, two sides and the included
angle, or two angles and any side of a triangle, the size and shape of the triangle
are determined. We can use the Law of Cosines or the Law of Sines to find the
measures of the remaining parts of the triangle.
Law of Cosines:
a2 5 b2 1 c2 2 2bc cos A
b2 5 a2 1 c2 2 2ac cos B
c2 5 a2 1 b2 2 2ab cos C
2
c2 2 a2
cos A 5 b 1 2bc
2
c2 2 b2
cos B 5 a 1 2ac
2
b2 2 c2
cos C 5 a 1 2ab
The Law of Cosines can be used to find the measure of the third side of a
triangle when the measures of two sides and the included angle are known. The
Law of Cosines can also be used to find the measure of any angle of a triangle
when the measures of three sides are known.
Law of Sines:
a
sin A
5 sinb B 5 sinc C
The Law of Sines can be used to find the measure of a side of a triangle
when the measures of any side and any two angles are known. The Law of Sines
can also be used to find the number of possible triangles that can be drawn
when the measures of two sides and an angle opposite one of them are known
and can be used to determine the measures of the remaining side and angles if
one or two triangles are possible.
Area of a Triangle:
Area of ABC 5 12bc sin A 5 12ac sin B 5 12ab sin C
The area of any triangle can be found if we know the measures of two sides
and the included angle.
An angle of elevation is an angle between a horizontal line and a line that
is rotated upward from the horizontal position. An angle of depression is an
angle between a horizontal line and a line that is rotated downward from the
horizontal position.
To determine the number of solutions given a, b, and mA in ABC, use
the Law of Sines to solve for sin B.
• If sin B . 1, there is no triangle.
• If sin B 5 1, there is one right triangle if A is acute but no triangle if A
is obtuse.
• If A is acute and sin B , 1, find two possible values of B:
0 , mB , 90 and mB9 5 180 2 mB.
If mA 1 mB9 , 180, there are
two possible triangles, ABC
and AB9C.
If mA 1 mB9 $ 180, AB9C is
not a triangle. There is only
one possible triangle, ABC.
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Trigonometric Applications
• If A is obtuse and sin B , 1, B must be acute: 0 , mB , 90.
If mA 1 mB , 180, there is
one triangle, ABC.
If mA 1 mB $ 180, there is no
triangle.
Alternatively, if we let h 5 b sin A, the height of the triangle, we can summarize the number of possible triangles given a, b, and mA in ABC:
C
C
C
C
b
h
b
A
A
A is:
Possible
triangles:
b
ha
Acute
a,h
None
B
b
a h a
B
A
Acute
h5a
One,
right B9
C
C
a
B
Acute
h,a,b
Two
h
a
A
h
B
Acute
a.b
One
a
b
B
a
h b
A
A
Obtuse
a,h
None
B
Obtuse
a.b
One
VOCABULARY
14-2 Law of Cosines
14-5 Law of Sines
14-6 Ambiguous case
14-7 Angle of elevation • Angle of Depression
REVIEW EXERCISES
1. In RST, RS 5 18, RT 5 27, and mR 5 50. Find ST to the nearest
integer.
2. The measures of two sides of a triangle are 12.0 inches and 15.0 inches.
The measure of the angle included between these two sides is 80 degrees.
Find the measure of the third side of the triangle to the nearest tenth of
an inch.
3. In DEF, DE 5 84, EF 5 76, and DF 5 94. Find, to the nearest degree,
the measure of the smallest angle of the triangle.
4. The measures of three sides of a triangle are 22, 46, and 58. Find, to the
nearest degree, the measure of the largest angle of the triangle.
5. Use the Law of Cosines to show that if the measures of the sides of a triangle are 10, 24, and 26, the triangle is a right triangle.
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Review Exercises
583
6. In ABC, AB 5 24, AC 5 40, and mA 5 30.
a. Find the area of ABC.
b. Find the length of the altitude from C to AB.
7. The lengths of the sides of a triangle are 8, 11, and 15.
a. Find the measure of the smallest angle of the triangle to the nearest
tenth.
b. Find the area of the triangle to the nearest tenth.
8. In ABC, BC 5 30.0, mA 5 70, and mB 5 55. Find, to the nearest
tenth, AB and AC.
9. The measures of two angles of a triangle are 100° and 46°. The length of
the shortest sides of the triangle is 12. Find, to the nearest integer, the
lengths of the other two sides.
10. In ABC, a 5 14, b 5 16, and mA 5 48.
a. How many different triangles are possible?
b. Find the measures of B and of C if ABC is an acute triangle.
c. Find the measures of B and of C if ABC is an obtuse triangle.
11. Show that it is not possible to draw PQR with p 5 12, r 5 15, and
mP 5 66.
12. The measure of a side of a rhombus is 28.0 inches and the measure of the
longer diagonal is 50.1 inches.
a. Find, to the nearest degree, the measure of each angle of the rhombus.
b. Find, to the nearest tenth, the measure of the shorter diagonal.
c. Find, to the nearest integer, the area of the rhombus.
13. Use the Law of Cosines to find two possible lengths for AB of ABC if
BC 5 7, AC 5 8, and mA 5 60.
14. Use the Law of Sines to show that there are two possible triangles if
BC 5 7, AC 5 8, and mA 5 60.
15. A vertical pole is braced by two wires that
extend from different points on the pole to the
same point on level ground. One wire is fastened to the pole 5.0 feet from the top of the
pole and makes an angle of 61 degrees with
the ground. The second wire is fastened to the
top of the pole and makes an angle of 66
degrees with the ground. Find the height of
the pole to the nearest tenth.
5 ft
66°
61°
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Trigonometric Applications
16. Coastguard station A is 12 miles west of
coastguard station B along a straight coastline. A ship is sited by the crew at station A
to the northeast of the station at an angle of
35 degrees with the coastline and by the
crew at station B to the northwest of the
station at an angle of 46 degrees with the
coastline. Find, to the nearest tenth, the distance from the ship to each of the stations.
35°
A
12 mi
46° B
D
17. In the diagram, ABC is a right triangle with
the right angle at C, AC 5 4.0, BC 5 3.0, and
mBAC 5 u. Side CB is extended to D and
mDAC 5 2u.
a. Find the exact value of sin 2u. (Hint: Use
the double-angle formulas.)
b. Find u to the nearest degree.
B
3.0
c. Find DC to the nearest tenth.
C 4.0 A
Exploration
Part A
Use software to draw any triangle, ABC.
Trisect each angle of the triangle.
Let D be the intersection of the trisection
lines from A and B that are closest to AB.
Let E be the intersection of the trisection
lines from B and C that are closest to BC.
Let F be the intersection of the trisection
lines from C and A that are closest to CA.
Measure DE, EF, and FD. What seems to
be true about DEF?
A
F
D
E
B
C
Part B
1. Draw any triangle, ABC. Measure the lengths of the sides and the measures of the angles.
2. Draw the trisection lines and label DEF as in part A.
3. Use the Law of Sines to find AD and BD in ABD.
4. Use the Law of Sines to find BE and CE in BCE.
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Cumulative Review
585
5. Use the Law of Sines to find CF and AF in CAF.
6. Use the Law of Cosines to find DF in ADF.
7. Use the Law of Cosines to find DE in BDE.
8. Use the Law of Cosines to find EF in CEF.
9. Is DEF an equilateral triangle?
CUMULATIVE REVIEW
CHAPTERS 1–14
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. The sum of !225 and 2 !216 is
(1) 213
(2) 13i
2. If f(u) 5 cot u, then f A p2 B is
(1) 0
(2) 1
(3) 218
(4) 18i
(3) `
(4) undefined
(3) 18
(4) 29
3
3. The expression a (k 1 1) 2 is equal to
k51
(1) 13
(2) 14
2
4. If f(x) 5 x 2 1 and g(x) 5 x , then f(g(3)) is equal to
(1) 18
(2) 11
(3) 8
(4) 4
5. The sum sin 50° cos 30° 1 cos 50° sin 30° is equal to
(1) sin 80°
(2) sin 20°
(3) cos 80°
(4) cos 20°
6. If
logx 14
5 –2, then x equals
(1) 16
(3) 21
(2) 2
1
(4) 16
7. The solution set of x2 2 2x 1 5 5 0 is
(1) {21, 3}
(3) {1 2 2i, 1 1 2i}
(2) {23, 1}
(4) {21 2 2i, 1 1 2i}
a
b
2
8. In simplest form, the fraction b1 1a is equal to
a2b
(1) a 1 b
(2) a – b
(3) 2(a 1 b)
9. An angle of 7p
4 radians is congruent to an angle of
(1) 135°
(2) 225°
(3) 315°
(4) b – a
(4) 405°
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Trigonometric Applications
10. Which of the following is a geometric sequence?
(1) 1, 3, 5, 7, . . .
(3) 1, 0.1, 0.01, 0.001, . . .
(2) 1, 3, 6, 10, . . .
(4) 1, 12, 13, 14, . . .
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. Use the exact values of the sin 30°, sin 45°, cos 30°, and cos 45° to find the
exact value of cos 15°.
!5
12. Write the fraction 11 2
as an equivalent fraction with a rational
1 !5
denominator.
Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
13. Solve for x and check: 3 – !2x 2 3 5 x
14. Find all values of x in the interval 0° # x , 360° that satisfy the following
equation:
6 sin2 x 2 5 sin x 2 4 5 0
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. If logb x 5 logb 3 1 2 logb 4 – 12 logb 8, express x in simplest form.
16. a. Sketch the graph of y 5 2 sin x in the interval 0 # x # 2p.
b. On the same set of axes, sketch the graph of y 5 cos 2x.
c. For how many values of x in the interval 0 # x # 2p does
2 sin x 5 cos 2x?
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CHAPTER
STATISTICS
Every year the admission officers of colleges
choose, from thousands of applicants, those students
who will be offered a place in the incoming class for
the next year. An attempt is made by the college to
choose students who will be best able to succeed academically and who best fit the profile of the student
body of that college. Although this choice is based not
only on academic standing, the scores on standardized
tests are an important part of the selection. Statistics
establishes the validity of the information obtained
from standardized tests and influence the interpretation of the data obtained from them.
15
CHAPTER
TABLE OF CONTENTS
15-1 Gathering Data
15-2 Measures of Central
Tendency
15-3 Measures of Central
Tendency for Grouped Data
15-4 Measures of Dispersion
15-5 Variance and Standard
Deviation
15-6 Normal Distribution
15-7 Bivariate Statistics
15-8 Correlation Coefficient
15-9 Non-Linear Regression
15-10 Interpolation and
Extrapolation
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
587
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Statistics
15-1 GATHERING DATA
Important choices in our lives are often made by evaluating information, but in
order to use information wisely, it is necessary to organize and condense the
multitude of facts and figures that can be collected. Statistics is the science that
deals with the collection, organization, summarization, and interpretation of
related information called data. Univariate statistics consists of one number for
each data value.
Collection of Data
Where do data come from? Individuals, government organizations, businesses,
and other political, scientific, and social groups usually keep records of their
activities. These records provide factual data. In addition to factual data, the outcome of an event such as an election, the sale of a product, or the success of a
movie often depends on the opinion or choices of the public.
Common methods of collecting data include the following:
1. Censuses: every ten years, the government conducts a census to determine
the U.S. population. Each year, almanacs are published that summarize
and update data of general interest.
2. Surveys: written questionnaires, personal interviews, or telephone requests
for information can be used when experience, preference, or opinions are
sought.
3. Controlled experiments: a structured study that usually consists of two
groups: one that makes use of the subject of the study (for example, a new
medicine) and a control group that does not. Comparison of results for the
two groups is used to indicate effectiveness.
4. Observational studies: similar to controlled experiments except that the
researcher does not apply the treatment to the subjects. For example, to
determine if a new drug causes cancer, it would be unethical to give the
drug to patients. A researcher observes the occurrence of cancer among
groups of people who previously took the drug.
When information is gathered, it may include data for all cases to which
the result of the study is to be applied. This source of information is called the
population. When the entire population can be examined, the study is a census.
For example, a study on the age and number of accidents of every driver insured
by an auto insurance company would constitute a census if all of the company’s
records are included. However, when it is not possible to obtain information
from every case, a sample is used to determine data that may then be applied to
every case. For example, in order to determine the quality of their product, the
quality-control department of a business may study a sample of the product
being produced.
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Gathering Data
589
In order that the sample reflect the properties of the whole group, the following conditions should exist:
1. The sample must be representative of the group being studied.
2. The sample must be large enough to be effective.
3. The selection should be random or determined in such a way as to eliminate any bias.
If a new medicine being tested is proposed for use by people of all ages, of
different ethnic backgrounds, and for use by both men and women, then the
sample must be made up of people who represent these differences in sufficient
number to be effective. A political survey to be effective must include people of
different cultural, ethnic, financial, geographic, and political backgrounds.
Potential Pitfalls of Surveys
Surveys are a very common way of collecting data. However, if not done correctly, the results of the survey can be invalid. One potential problem is with the
wording of the survey questions. For example, the question, “Do you agree that
teachers should make more money?” will likely lead to a person answering
“Yes.” A more neutral form of this question would be, “Do you believe that
teachers’ salaries are too high, too low, or just about right?”
Questions can also be too vague, “loaded” (that is, use words with unintended connotations), or confusing. For example, for many people, words that
invoke race will likely lead to an emotional response.
Another potential problem with surveys is the way that participants are
selected. For example, a magazine would like to examine the typical teenager’s
opinion on a pop singer in a given city. The magazine editors conduct a survey
by going to a local mall. The problem with this survey is that teenagers who go
to the mall are not necessarily representative of all teenagers in the city. A better survey would be done by visiting the high schools of the city.
Many surveys often rely on volunteers. However, volunteers are likely to
have stronger opinions than the general population. This is why the selection of
participants, if possible, should be random.
EXAMPLE 1
A new medicine intended for use by adults is being tested on five men whose
ages are 22, 24, 25, 27, and 30. Does the sample provide a valid test?
Solution No:
• The sample is too small.
• The sample includes only men.
• The sample does not include adults over 30.
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Statistics
EXAMPLE 2
The management of a health club has received complaints about the temperature of the water in the swimming pool. They want to sample 50 of the 200 members of the club to determine if the temperature of the pool should be changed.
How should this sample be chosen?
Solution One suggestion might be to poll the first 50 people who use the pool on a
given day. However, this will only include people who use the pool and who
can therefore tolerate the water temperature.
Another suggestion might be to place 50 questionnaires at the entrance
desk and request members to respond. However, this includes only people
who choose to respond and who therefore may be more interested in a
change.
A third suggestion might be to contact by phone every fourth person on
the membership list and ask for a response. This method will produce a random
sample but will include people who have no interest in using the pool. This
sample may be improved by eliminating the responses of those people.
Organization of Data
In order to be more efficiently presented and more easily understood and interpreted, the data collected must be organized and summarized. Charts and
graphs such as the histogram are useful tools. The stem-and-leaf diagram is an
effective way of organizing small sets of data.
For example, the heights of
Heights of Children
the 20 children in a seventhgrade class are shown to the 61 71 58 72 60 53 74 61 68 65
right. To draw a stem-and-leaf 72 67 64 48 70 56 65 67 59 61
diagram, choose the tens digit as
the stem and the units digit as the
leaf.
(1) Draw a vertical line and list
the tens digits, 4, 5, 6, and 7
(or the stem), from bottom
to top to the left of the line:
Stem
(2) Enter each height by writing
the leaf, the units digit, to
the right of the line, following the appropriate stem:
Stem
7
6
5
4
7
6
5
4
Leaf
12420
1018574571
8369
8
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Gathering Data
(3) Arrange the leaves in numerical order after
each stem:
(4) Add a key to indicate the meaning of the
numbers in the diagram:
Stem
591
Leaf
7
01224
6
0111455778
5
3689
4
8
Key: 4 8 5 48
For larger sets of data, a frequency distribution
table can be drawn. Information is grouped and the
frequency, the number of times that a particular
value or group of values occurs, is stated for each
group. For example, the table to the right lists the
scores of 250 students on a test administered to all
tenth graders in a school district.
The table tells us that 24 students scored between
91 and 100, that 82 scored between 81 and 90, that 77
scored between 71 and 80. The table also tells us that
the largest number of students scored in the 80’s and
that ten students scored 50 or below. Note that unlike
the stem-and-leaf diagram, the table does not give us
the individual scores in each interval.
Score
Frequency
91–100
24
81–90
82
71–80
77
61–70
36
51–60
21
41–50
8
31–40
2
EXAMPLE 2
The prices of a gallon of milk in 15 stores are listed below.
$3.15 $3.39 $3.28 $2.98 $3.25 $3.45 $3.58 $3.24
$3.35 $3.29 $3.29 $3.30 $3.25 $3.40 $3.29
a. Organize the data in a stem-and-leaf diagram.
b. Display the data in a frequency distribution table.
c. If the 15 stores were chosen at random from the more than 100 stores that sell
milk in Monroe County, does the data set represent a population or a sample?
Solution a. Use the first two
digits of the price
as the stem.
Use the last digit as
the leaf.
Stem
Stem
3.5
3.4
3.3
3.2
3.1
3.0
2.9
3.5
3.4
3.3
3.2
3.1
3.0
2.9
Leaf
8
50
950
8549959
5
8
Write the leaves in
numerical order.
Stem
Leaf
3.5
8
3.4
05
3.3
059
3.2
4558999
3.1
5
3.0
2.9
8
Key: 2.9 8 5 2.98
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Statistics
b. Divide the data into groups of length $0.10 starting with $2.90. These groups
correspond with the stems of the stem-and-leaf diagram. The frequencies
can be determined by the use of a tally to represent each price.
Stem
Leaf
3.5
8
3.4
05
3.3
059
3.2
4558999
3.1
5
3.0
2.9
8
Key: 2.9 8 5 2.98
Price
Tally
$3.50–$3.59
|
1
$3.40–$3.49
||
2
$3.30–$3.39
|||
3
$3.20–$3.29
|||| ||
7
$3.10–$3.19
|
1
Frequency
$3.00–$3.09
0
$2.90–$2.99
|
1
c. The data set is obtained from a random selection of stores from all of
the stores in the study and is therefore a sample. Answer
Once the data have been organized, a graph can be used to visualize the
intervals and their frequencies. A histogram is a vertical bar graph where each
interval is represented by the width of the bar and the frequency of the interval
is represented by the height of the bar. The bars are placed next to each other
to show that, as one interval ends, the next interval begins. The histogram below
shows the data of Example 2:
PRICE OF A GALLON OF MILK
Frequency
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8
7
6
5
4
3
2
1
0
.99
.90
$2
.19
.09
2
–$
$3
.00
3
–$
$3
.10
3
–$
.20
$3
Price
.49
.39
.29
3
–$
$3
.30
3
–$
.40
$3
.59
3
–$
3
–$
.50
$3
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Gathering Data
593
A graphing calculator can be used to display a histogram from the data on
a frequency distribution table.
(1) Clear L1 and L2, the lists to be used, of existing data.
ENTER:
STAT
4
2nd
L1
,
2nd
1 to edit the lists. Enter the
(2) Press STAT
minimum value of each interval in L1 and
the frequencies into L2.
L2
ENTER
L1
L2
3.5
1
3.4
2
3.3
3
3.2
7
3.1
1
3
0
2.9
1
L3(1)=
L3
(3) Clear any functions in the Y menu.
(4) Turn on Plot1 from the STAT PLOT menu
and select for histogram. Make
sure to also set Xlist to L1 and Freq to L2.
ENTER:
2nd
STAT PLOT
䉲
䉴
䉴
L1
䉲
2nd
ENTER
Plot1 Plot2 Plot3
On
Off
.
Ty p e : ......
ENTER
1
䉲
2nd
Xlist:L1
Freq:L2
L2
(5) In the WINDOW menu, enter Xmin as 2.8,
the length of one interval less than the
smallest interval value, and Xmax as 3.7, the
length of one interval more than the largest
interval value. Enter Xscl as 0.10, the
length of the interval. The Ymin is 0 and
Ymax is 9 to be greater than the largest
frequency.
(6) Press GRAPH to graph the histogram.
We can view the frequency associated
with each interval by pressing TRACE .
Use the left and right arrows to move
from one interval to the next.
WINDOW
Xmin =2.8
Xmax=3.7
Xscl =.1
Ymin =0
Ymax=9
Yscl =1
Xres=1
3
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Statistics
Exercises
Writing About Mathematics
1. In a controlled experiment, two groups are formed to determine the effectiveness of a new
cold remedy. One group takes the medicine and one does not. Explain why the two groups
are necessary.
2. In the experiment described in Exercise 1, explain why it is necessary that a participant does
not know to which group he or she belongs.
Developing Skills
In 3–5, organize the data in a stem-and-leaf diagram.
3. The grades on a chemistry test:
95 90 84 85 74 67 78 86 54 82
75 67 92 66 90 68 88 85 76 87
4. The weights of people starting a weight-loss program:
173 210 182 190 175 169 236 192 203 196 201
187 205 195 224 177 195 207 188 184 196 155
5. The heights, in centimeters, of 25 ten-year-old children:
137 134 130 144 131 141 136 140 137 129 139 137 144
127 147 143 132 132 142 142 131 129 138 151 137
In 6–8, organize the data in a frequency distribution table.
6. The numbers of books read during the summer months by each of 25 students:
2 2 5 1 3 0 7 2 4 3 3 1 8
5 7 3 4 1 0 6 3 4 1 1 2
7. The sizes of 26 pairs of jeans sold during a recent sale:
8 12 14 10 12 16 14 6 10 9 8 13 12
8 12 10 12 14 10 12 16 10 11 15 8 14
8. The number of siblings of each of 30 students in a class:
2 1 1 5 1 0 2 2 1 3 4 0 6 2 0
3 1 2 2 1 1 1 0 2 1 0 1 1 2 3
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595
Gathering Data
In 9–11, graph the histogram of each set of data.
9.
10.
xi
fi
xi
35–39
13
101–110
30–34
19
25–29
fi
11.
xi
fi
3
$55–$59
20
91–100
6
$50–$54
15
10
81–90
10
$45–$49
12
20–24
13
71–80
13
$40–$44
5
15–19
8
61–70
14
$35–$39
10
10–14
19
51–60
2
$30–$34
12
5–9
15
41–50
2
$25–$29
16
Applying Skills
In 12–18, suggest a method that might be used to collect data for each study. Tell whether your
method uses a population or a sample.
12. Average temperature for each month for a given city
13. Customer satisfaction at a restaurant
14. Temperature of a patient in a hospital over a period of time
15. Grades for students on a test
16. Population of each of the states of the United States
17. Heights of children entering kindergarten
18. Popularity of a new movie
19. The grades on a math test of 25 students are listed below.
86 92 77 84 75 95 66 88 84 53 98 87 83
74 61 82 93 98 87 77 86 58 72 76 89
a. Organize the data in a stem-and-leaf diagram.
b. Organize the data in a frequency distribution table.
c. How many students scored 70 or above on the test?
d. How many students scored 60 or below on the test?
20. The stem-and-leaf diagram at the right shows the ages of 30 people
in an exercise class. Use the diagram to answer the following questions.
a. How many people are 45 years old?
b. How many people are older than 60?
c. How many people are younger than 30?
d. What is the age of the oldest person in the class?
e. What is the age of the youngest person in the class?
Stem
Leaf
7
2
6
015
5
136679
4
2245567
3
9
2
1335788
1
02269
Key: 1 9 5 19
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Hands-On Activity
In this activity, you will take a survey of 25 people. You will need a stopwatch or a clock with a second hand. Perform the following experiment with each participant to determine how each perceives
the length of a minute:
1. Indicate the starting point of the minute.
2. Have the person tell you when he or she believes that a minute has passed.
3. Record the actual number of seconds that have passed.
After surveying all 25 participants, use a stem-and-leaf diagram to record your data. Keep this
data. You will use it throughout your study of this chapter.
15-2 MEASURES OF CENTRAL TENDENCY
After data have been collected, it is often useful to represent the data by a single value that in some way seems to represent all of the data. This number is
called the measure of central tendency. The most frequently used measures of
central tendency are the mean, the median, and the mode.
The Mean
The mean or arithmetic mean is the most common measure of central tendency. The mean is the sum of all of the data values divided by the number of
data values.
For example, nine members of the basketball team played during all or part
of the last game. The number of points scored by each of the players was:
21, 15, 12, 9, 8, 7, 5, 2, 2
Mean 5 21 1 15 1 12 1 9 19 8 1 7 1 5 1 2 1 2 5 81
9 59
Note that if each of the 9 players had scored 9 points, the total number of
points scored in the game would have been the same.
The summation symbol, S, is often used to designate the sum of the data values. We designate a data value as xi and the sum of n data values as
n
c1x
n
a xi 5 x1 1 x2 1 x3 1
i51
For the set of data given above:
n 5 9, x1 5 21, x2 5 15, x3 5 12, x4 5 9, x5 5 8, x6 5 7, x7 5 5, x8 5 2, x9 5 2
9
a xi 5 81
i51
The subscript for each data value indicates its position in a list of data values, not its value, although the value of i and the data value may be the same.
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Measures of Central Tendency
Procedure
To find the mean of a set of data:
1. Add the data values.
2. Divide the sum by n, the total number of data values.
EXAMPLE 1
An English teacher recorded the number of spelling errors in the 40 essays written by students. The table below shows the number of spelling errors and the
frequency of that number of errors, that is, the number of essays that contained
that number of misspellings. Find the mean number of spelling errors for these
essays.
Errors
0
1
2
3
4
5
6
7
8
9
10
Frequency
1
3
2
2
6
9
7
5
2
1
2
Solution To find the total number of spelling errors, first multiply each number of
errors by the frequency with which that number of errors occurred. For example, since 2 essays each contained 10 errors, there were 20 errors in these
essays. Add the products in the fi xi row to find the total number of errors in
the essays. Divide this total by the total frequency.
Total
Errors (xi)
0
1
2
3
4
5
6
7
8
9
10
Frequency (fi)
1
3
2
2
6
9
7
5
2
1
2
40
Errors Frequency (fixi)
0
3
4
6
24
45
42
35
16
9
20
204
10
a fixi
Mean 5
i50
40
5 204
40 5 5.1
Note that for this set of data, the data value is equal to i for each xi.
Answer 5.1 errors
The Median
The median is the middle number of a data set arranged in numerical order.
When the data are arranged in numerical order, the number of values less than
or equal to the median is equal to the number of values greater than or equal to
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Statistics
the median. Consider again the nine members of the basketball team who
played during all or part of the last game and scored the following number of
points:
2, 2, 5, 7, 8, 9, 12, 15, 21
We can write 9 5 4 1 1 1 4. Therefore, we think of four scores below the
median, the median, and four scores above the median.
8,
scores less than the median
↑
9, 12, 15, 21
d
2, 2, 5, 7,
scores greater than the median
median
Note that the number of data values can be written as 2(4) 1 1. The median
is the (4 1 1) or 5th value from either end of the distribution. The median number of points scored is 8.
When the number of values in a set of data is even, then the median is the
mean of the two middle values. For example, eight members of a basketball
team played in a game and scored the following numbers of points.
4, 6, 6, 7, 11, 12, 18, 20
We can separate the eight data values into two groups of 4 values. Therefore,
we average the largest score of the four lowest scores and the smallest score of
the four highest scores.
4, 6, 6, 7,
four lowest scores
7 1 11
2
↑
,
11, 12, 18, 20
d
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four highest scores
median
median 5 7 12 11 5 9
Note that the number of data values can be written as 2(4). The median is
the mean of the 4th and 5th value from either end of the distribution. The
median is 9. There are four scores greater than the median and four scores lower
than the median. The median is a middle mark.
Procedure
To find the median of a set of data:
1. Arrange the data in order from largest to smallest or from smallest to
largest.
2. a. If the number of data values is odd, write that number as 2n 1 1.The
median is the score that is the (n 1 1)th score from either end of the
distribution.
b. If the number of data values is even, write that number as 2n.The median
is the score that is the mean of the nth score and the (n 1 1)th score
from either end of the distribution.
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Measures of Central Tendency
599
The Mode
The mode is the value or values that occur most frequently in a set of data. For
example, in the set of numbers
3, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 10,
the number that occurs most frequently is 6. Therefore, 6 is the mode for this set
of data.
When a set of numbers is arranged in a
Number of
frequency distribution table, the mode is the
Books Read
Frequency
entry with the highest frequency. The table to
8
1
the right shows, for a given month, the number of books read by each student in a class.
7
1
The largest number of students, 12, each
6
3
read four books. The mode for this distribu5
6
tion is 4.
A data set may have more than one
4
12
mode. For example, in the set of numbers 3, 4,
3
4
5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 10, the numbers 5 and
2
2
6 each occur four times, more frequently than
any other number. Therefore, 5 and 6 are
1
0
modes for this set of data. The set of data is
0
1
said to be bimodal.
Quartiles
When a set of data is listed in numerical order, the median separates the data
into two equal parts. The quartiles separate the data into four equal parts. To
find the quartiles, we first separate the data into two equal parts and then separate each of these parts into two equal parts. For example, the grades of 20 students on a math test are listed below.
58, 60, 65, 70, 72, 75, 76, 80, 80, 81, 83, 84, 85, 87, 88, 88, 90, 93, 95, 98
Since there are 20 or 2(10) grades, the median grade that separates the data
into two equal parts is the average of the 10th and 11th grade.
58, 60, 65, 70, 72, 75, 76, 80, 80, 81,
Lower half
83, 84, 85, 87, 88, 88, 90, 93, 95, 98
↑
82
i
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Upper half
Median
The 10th grade is 81 and the 11th grade is 83. Therefore, the mean of these
83
two grades, 81 1
or 82, separates the data into two equal parts. This number
2
is the median grade.
Page 600
Statistics
Now separate each half into two equal parts. Find the median of the two
lower quarters and the median of the two upper quarters.
D
D
↑
73.5
D
58, 60, 65, 70, 72 75, 76, 80, 80, 81 83, 84, 85, 87, 88 88, 90, 93, 95, 98
↑
82
↑
88
The numbers 73.5, 82, and 88 are the quartiles for this data.
• One quarter of the grades are less than or equal to 73.5. Therefore, 73.5 is
the first quartile or the lower quartile.
• Two quarters of the grades are less than or equal to 82. Therefore, 82 is the
second quartile. The second quartile is always the median.
• Three quarters of the grades are less than or equal to 88. Therefore, 88 is
the third quartile or the upper quartile.
Note: The minimum, first quartile, median, third quartile, and maximum make
up the five statistical summary of a data set.
When the data set has an odd number of values, the median or second quartile will be one of the values. This number is not included in either half of the
data when finding the first and third quartiles.
For example, the heights, in inches, of 19 children are given below:
37, 39, 40, 42, 42, 43, 44, 44, 44, 45, 46, 47, 47, 48, 49, 49, 50, 52, 53
There are 19 5 2(9) 1 1 data values. Therefore, the second quartile is the
10th height or 45.
45 , 46, 47, 47, 48, 49, 49, 50, 52, 53
I
37, 39, 40, 42, 42, 43, 44, 44, 44,
I
There are 9 5 2(4) 1 1 heights in the lower half of the data and also in the
upper half of the data. The middle height of each half is the 5th height.
45 , 46, 47, 47, 48,
49 , 49, 50, 52, 53
D
42 , 43, 44, 44, 44,
D
37, 39, 40, 42,
D
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↑
↑
↑
Lower quartile
Median
Upper quartile
In this example, the first or lower quartile is 42, the median or second quartile is 45, and the third or upper quartile is 49. Each of these values is one of the
data values, and the remaining values are separated into four groups with the
same number of heights in each group.
Box-and-Whisker Plot
A box-and-whisker plot is a diagram that is used to display the quartile values
and the maximum and minimum values of a distribution. We will use the data
from the set of heights given above.
Page 601
Measures of Central Tendency
45 , 46, 47, 47, 48,
601
49 , 49, 50, 52, 53
D
42 , 43, 44, 44, 44,
37, 39, 40, 42,
D
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1. Choose a scale that includes the maximum and minimum values of the
data. We will use a scale from 35 to 55.
2. Above the scale, place dots to represent the minimum value, the lower
quartile, the mean, the upper quartile, and the maximum value.
35
42
37
45
49
53
55
3. Draw a box with opposite sides through the lower and upper quartiles and
a vertical line through the median.
35
42
37
45
49
53
55
4. Draw whiskers by drawing a line to join the dot that represents the minimum value to the dot that represents the lower quartile and a line to join
the dot that represents the upper quartile to the dot that represents the
maximum value.
35
42
37
45
49
53
55
A graphing calculator can be used to find the quartiles and to display the
box-and-whisker plot.
(1) Enter the data for this set of heights into L1.
(2) Turn off all plots and enter the required
choices in Plot 1.
ENTER:
䉲
2nd
2nd
STAT PLOT
䉴
L1
䉴
䉴
䉲
ALPHA
On
Off
.
Ty p e : ......
ENTER
1
䉴
Plot1 Plot2 Plot3
ENTER
1
䉲
Xlist:L1
Freq:1
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Statistics
(3) Now display the plot by entering
9 . You can press TRACE
ZOOM
to
display the five statistical summary.
The five statistical summary of a set of data
can also be displayed on the calculator by using
1-Var Stats.
ENTER:
STAT
䉴
ENTER
ENTER
The first entry under 1-Var Stats is –x, the value of the mean. The next is
a x, the sum of the data values. The next three entries are values that will be
used in the sections that follow. The last entry is the number of data values.
The arrow tells us that there is more information. Scroll down to display the
minimum value, minX 5 37, the lower quartile, Q1 5 42, the median or second quartile, Med 5 45, the upper quartile, Q3 5 49, and the maximum value,
maxX 5 53.
>
1–Var Stats
–
x=45.31578947
x=861
x2=39353
Sx=4.321170515
X=4.205918531
n=19
1–Var Stats
n=19
minX=37
Q1=42
Med=45
Q3=49
maxX=53
>
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EXAMPLE 1
Find the mean, the median, and the mode of the following set of grades:
92, 90, 90, 90, 88, 87, 85, 70
n
Solution
Mean 5 n1 a xi 5 18(92 1 90 1 90 1 90 1 88 1 87 1 85 1 70) 5 692
8 5 86.5
i51
88
Median 5 the average of the 4th and 5th grades 5 90 1
5 89
2
Mode 5 the grade that appears most frequently 5 90
EXAMPLE 2
The following list shows the length of time, in minutes, for each of 35 employees
to commute to work.
25 12 20 18 35 25 40 35 27 30 60 22 36 20 18
27 35 42 35 55 27 30 15 22 10 35 27 15 57 18
25 45 24 27 25
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Measures of Central Tendency
603
a. Organize the data in a stem-and-leaf diagram.
b. Find the median, lower quartile, and upper quartile.
c. Draw a box-and-whisker plot.
Solution a. (1) Choose the tens digit as the
stem and enter the units digit
as the leaf for each value.
Stem
(2) Write the leaves in numerical
order from smallest to largest.
Stem
b. (1) The median is the middle value
of the 35 data values when the
values are arranged in order.
35 5 2(17) 1 1
The median is the 18th value.
Separate the data into groups
of 17 from each end of the distribution. The median is 27.
(2) There are 17 5 2(8) 1 1 data
values below the median. The
lower quartile is the 9th data
value from the lower end.
The upper quartile is the 9th
data value from the upper
end.
6
5
4
3
2
1
Leaf
0
57
025
55065505
50572077275475
2885058
Leaf
6
0
5
57
4
025
3
00555556
2
00224555577777
1
0255888
Key: 1 0 5 10
Stem
6
5
4
3
2
1
Stem
6
5
4
3
2
1
Leaf
0
57
025
005 5 5556
002 2 455557 7 777
0255888
Leaf
0
57
025
005 5 5 5 56
0 0 2 2 455557 7 777
0255888
The lower quartile is 20, the
median is 27, and the upper quartile is 35.
c.
10
20
27
35
60
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Statistics
Note: In the stem-and-leaf diagram of Example 2, the list is read from left to
right to find the lower quartile. The list is read from right to left (starting from
the top) to find the upper quartile.
Exercises
Writing About Mathematics
1. Cameron said that the number of data values of any set of data that are less than the lower
quartile or greater than the upper quartile is exactly 50% of the number of data values. Do
you agree with Cameron? Explain why or why not.
2. Carlos said that for a set of 2n data values or of 2n 1 1 data values, the lower quartile is the
median of the smallest n values and the upper quartile is the median of the largest n values.
Do you agree with Carlos? Explain why or why not.
Developing Skills
In 3–8, find the mean, the median, and the mode of each set of data.
3. Grades: 74, 78, 78, 80, 80, 80, 82, 88, 90
4. Heights: 60, 62, 63, 63, 64, 65, 66, 68, 68, 68, 70, 75
5. Weights: 110, 112, 113, 115, 115, 116, 118, 118, 125, 134, 145, 148
6. Number of student absences: 0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 5, 9
7. Hourly wages: $6.90, $7.10, $7.50, $7.50, $8.25, $9.30, $9.50, $10.00
8. Tips: $1.00, $1.50, $2.25, $3.00, $3.30, $3.50, $4.00, $4.75, $5.00, $5.00, $5.00
In 9–14, find the median and the first and third quartiles for each set of data values.
9. 2, 3, 5, 8, 9, 11, 15, 16, 17, 20, 22, 23, 25
10. 34, 35, 35, 36, 38, 40, 42, 43, 43, 43, 44, 46, 48, 50
11. 23, 27, 15, 38, 12, 17, 22, 39, 28, 20, 27, 18, 25, 28, 30, 29
12. 92, 86, 77, 85, 88, 90, 81, 83, 95, 76, 65, 88, 91, 81, 88, 87, 95
13. 75, 72, 69, 68, 66, 65, 64, 63, 63, 61, 60, 59, 59, 58, 56, 54, 52, 50
14. 32, 32, 30, 30, 29, 27, 26, 22, 20, 20, 19, 18, 17
15. A student received the following grades on six tests: 90, 92, 92, 95, 95, x.
a. For what value(s) of x will the set of grades have no mode?
b. For what value(s) of x will the set of grades have only one mode?
c. For what value(s) of x will the set of grades be bimodal?
16. What are the first, second, and third quartiles for the set of integers from 1 to 100?
17. What are the first, second, and third quartiles for the set of integers from 0 to 100?
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Measures of Central Tendency for Grouped Data
Applying Skills
18. The grades on a English test are shown in the
stem-and-leaf diagram to the right.
a. Find the mean grade.
b. Find the median grade.
c. Find the first and third quartiles.
d. Draw a box-and-whisker plot for this data.
Stem
605
Leaf
9
001259
8
022555788
7
35667
6
055
5
5
4
7
Key: 4 7 5 47
19. The weights in pounds of the members of the football team are shown below:
181 199 178 203 211 208 209 202 212 194
185 208 223 206 202 213 202 186 189 203
a. Find the mean.
b. Find the median.
c. Find the mode or modes.
d. Find the first and third quartiles.
e. Draw a box-and-whisker plot.
20. Mrs. Gillis gave a test to her two classes of algebra. The mean grade for her class of 20 students was 86 and the mean grade of her class of 15 students was 79. What is the mean grade
when she combines the grades of both classes?
Hands-On Activity
Use the estimates of a minute collected in the Hands-On Activity of the previous section to determine the five statistical summary for your data. Draw a box-and-whisker plot to display the data.
15-3 MEASURES OF CENTRAL TENDENCY FOR GROUPED DATA
Most statistical studies involve much larger numbers of data values than can be
conveniently displayed in a list showing each data value. Large sets of data are
usually organized into a frequency distribution table.
Frequency Distribution Tables for
Individual Data Values
A frequency distribution table records the individual data values and the frequency or number of times that the data value occurs in the data set. The example on page 606 illustrates this method of recording data.
Each of an English teacher’s 100 students recently completed a book report.
The teacher recorded the number of misspelled words in each report. The table
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Statistics
records the number of reports for each number of misspelled words. Let xi represent the number of misspelled words in a report and fi represent the number
of reports that contain xi misspelled words.
Total
xi
0
1
2
3
4
5
6
7
8
9
10
fi
5
7
6
8
19
26
16
7
5
0
1
100
xifi
0
7
12
24
76
130
96
49
40
0
10
444
The mean of this set of data is the total number of misspelled words divided
by the number of reports. To find the total number of misspelled words, we must
first multiply each number of misspelled words, xi, by the number of reports that
contain that number of misspelled words, fi. That is, we must find xi fi for each
number of misspelled words.
10
For this set of data, when we sum the fi row, a fi 5 100, and when we sum
i50
10
the xi fi row a xifi 5 444.
i50
10
a xi fi
Mean 5
i50
10
a fi
5 444
100 5 4.44
i50
To find the median and the quartiles for this set of data, we will find the cumulative frequency for each number of misspelled words. The cumulative frequency
is the accumulation or the sum of all frequencies less than or equal to a given
frequency. For example, the cumulative frequency for 3 misspelled words on a
report is the sum of the frequencies for 3 or fewer misspelled words, and the
cumulative frequency for 6 misspelled words on a report is the sum of the frequencies for 6 or fewer misspelled words. The third row of the table shows that
0 misspelled words occur 5 times, 1 or fewer occur 7 1 5 or 12 times, 2 or fewer
occur 6 1 12 or 18 times. In each case, the cumulative frequency for xi is the frequency for xi plus the cumulative frequency for xi21. The cumulative frequency
for the largest data value is always equal to the total number of data values.
xi
0
1
2
3
4
5
6
7
8
9
10
fi
5
7
6
8
19
26
16
7
5
0
1
Cumulative
Frequency
5
12
18
26
45
71
87
94
99
99
100
↑
3
Lower
quartile
↑
5
↑
6
Median Upper
quartile
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Measures of Central Tendency for Grouped Data
607
Since there are 100 data values, the median is the average of the 50th and
51st values. To find these values, look at the cumulative frequency column. There
are 45 values less than or equal to 4 and 71 less than or equal to 5. Therefore,
the 50th and 51st values are both 5 and the median is 5 misspelled words.
Similarly, the upper quartile is the average of the 75th and 76th values. Since
there are 71 values less than or equal to 5, the 75th and 76th values are both 6
misspelled words. The lower quartile is the average of the 25th and 26th values.
Since there are 18 values less than or equal to 2, the 25th and 26th values are
both 3 misspelled words.
Percentiles
A percentile is a number that tells us what percent of the total number of data
values lie at or below a given measure.
For example, let us use the data from the previous section. The table records
the number of reports, fi, that contain each number of misspelled words, xi , on
100 essays.
xi
0
1
2
3
4
5
6
7
8
9
10
fi
5
7
6
8
19
26
16
7
5
0
1
Cumulative
Frequency
5
12
18
26
45
71
87
94
99
99
100
To find the percentile rank of 7 misspelled words, first find the number of
essays with fewer than 7 misspelled words, 87. Add to this half of the essays with
7 misspelled words, 72 or 3.5. Add these two numbers and divide the sum by the
number of essays, 100.
87 1 3.5
100
5 90.5
100 5 90.5%
Percentiles are usually not written with fractions. We say that 7 misspelled
words is at the 90.5th or 91st percentile. That is, 91% of the essays had 7 or fewer
misspelled words.
Frequency Distribution Tables for Grouped Data
Often the number of different data values in a set of data is too large to list each
data value separately in a frequency distribution table. In this case, it is useful to
list the data in terms of groups of data values rather than in terms of individual
data values. The following example illustrates this method of recording data.
There are 50 members of a weight-loss program. The weights range from
181 to 285 pounds. It is convenient to arrange these weights in groups of 10
pounds starting with 180–189 and ending with 280–289. The frequency distribution table shows the frequencies of the weights for each interval.
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Statistics
Weights
Midpoint
xi
Frequency
fi
280–289
270–279
260–269
250–259
240–249
230–239
220–229
210–219
200–209
190–199
180–189
284.5
274.5
264.5
254.5
244.5
234.5
224.5
214.5
204.5
194.5
184.5
1
3
4
8
12
10
5
3
2
1
1
Cumulative
Frequency
xifi
284.5
823.5
1,058.0
2,036.0
2,934.0
2,345.0
1,122.5
643.5
409.0
194.5
184.5
50
49
46
42
34
22
12
7
4
2
1
In order to find the mean, assume that the
weights are evenly distributed throughout the
interval. The mean is found by using the midpoint of the weight intervals as representative of
each value in the interval groupings.
11
a xi fi
Mean 5
i51
11
a fi
12,035
5 50
5 240.7
i51
To find the median for a set of data that is
organized in intervals greater than 1, first find
50
12,035
the interval in which the median lies by using
the cumulative frequency.
There are 50 data values. Therefore, the median is the value between the
25th and the 26th values. The cumulative frequency tells us that there are 22 values less than or equal to 239 and 34 values less than or equal to 249. Therefore,
the 25th and 26th values are in the interval 240–249. Can we give a better
approximation for the median?
The endpoints of an interval are the lowest and highest data values to be
entered in that interval. The boundary values are the values that separate intervals. The lower boundary of an interval is midway between the lower endpoint
of the interval and the upper endpoint of the interval that precedes it. The lower
boundary of the 240–249 is midway between 240 and 239, that is, 239.5. The
upper endpoint of this interval is between 249 and 250, that is 249.5.
Since there are 34 weights less than or equal to 249 and 22 weights less than
240, the weights in the 240–249 interval are the 23rd through the 34th weights.
Think of these 12 weights as being evenly spaced throughout the interval.
median
23
24
25
26
27
29
28
30
31
32
239.5
33
34
249.5
3
12
3
The midpoint between the 25th and 26th weights is 12
of the distance
between the boundaries of the interval, a difference of 10.
3
Median 5 239.5 1 12
(10) 5 239.5 1 2.5 5 242
The estimated median for the weights is 242 pounds. Thus, if we assume that
the data values are evenly distributed within each interval, we can obtain a better approximation for the median.
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Measures of Central Tendency for Grouped Data
609
EXAMPLE 1
The numbers of pets owned by the children in a
sixth-grade class are given in the table.
No. of Pets
6
5
4
3
2
1
0
a. Find the mean.
b. Find the median.
c. Find the mode for this set of data.
d. Find the percentile rank of 4 pets.
Frequency
2
1
3
5
8
14
7
Solution a. Add to the table the number of pets times the frequency, and the cumulative frequency for each row of the table.
No. of Pets
xi
Frequency
fi
fi xi
Cumulative
Frequency
6
5
4
3
2
1
0
2
1
3
5
8
14
7
12
5
12
15
16
14
0
40
38
37
34
29
21
7
40
74
Add the numbers in the fixi column and the numbers in the fi column.
6
6
a fi 5 40
a fixi 5 74
i50
i50
6
a xi fi
Mean 5
i50
6
a fi
5 74
40 5 1.85
i50
b. There are 40 data values in this set of data. The median is the average of the
20th and the 21st data values. Since there are 21 students who own 1 or
fewer pets, both the 20th and the 21st data value is 1. Therefore, the median
number of pets is 1.
c. The largest number of students, 14, have 1 pet. The mode is 1.
d. There are 34 students with fewer than 4 pets and 3 students with 4 pets. Add
34 and half of 3 and divide the sum by the total number of students, 40.
34 1 23
40
5 35.5
40 5 0.8875 5 88.75%
Four pets represents the 89th percentile.
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Statistics
Calculator Clear the lists first if necessary by keying in STAT
Solution for
2nd
ENTER .
L2
a, b, and c
4
2nd
L2
ENTER
L1
,
Enter the number of pets in L1.
Enter the frequency for each number of pets in L2.
Display 1-Var Stats.
ENTER:
DISPLAY:
STAT
䉴
2nd
1
L1
,
2nd
1–Var Stats
–
x=1.85
x=74
x2=236
Sx=1.594059485
x=1.574007624
n=40
1–Var Stats
n=40
minX=0
Q1=1
Med=1
Q3=3
maxX=6
>
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>
– , is the mean, 1.85. Use the down-arrow key,
The first entry, x
the median, Med. The median is 1.
6
䉲
, to display
6
Note that a x 5 74 is a fixi and n 5 40 is a fi.
i50
i50
Answers a. mean 5 1.85 b. median 5 1 c. mode 5 1 d. 4 is the 89th percentile
Note: The other information displayed under 1-Var Stats will be used in the
sections that follow.
EXAMPLE 2
A local business made the summary of the ages of 45 employees shown below.
Find the mean and the median age of the employees to the nearest integer.
Age
Frequency
20–24 25–29 30–34 35–39 40–44 45–49
1
2
5
6
7
10
50–54 55–59 60–64
7
5
Solution Add to the table the midpoint, the midpoint times the frequency, and the
cumulative frequency for each interval.
2
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Measures of Central Tendency for Grouped Data
Age
Midpoint
(xi)
Frequency
(fi)
xifi
Cumulative
Frequency
60–64
55–59
50–54
45–49
40–44
35–39
30–34
25–29
20–24
62
57
52
47
42
37
32
27
22
2
5
7
10
7
6
5
2
1
124
285
364
470
294
222
160
54
22
45
43
38
31
21
14
8
3
1
45
1,995
611
11
a xi fi
Mean 5
i51
11
a fi
1,995
5 45 5 44.333 . . . 44
i51
The mean is the middle age of 45 ages or the 23rd age. Since there are 21 ages
less than 45, the 23rd age is the second of 10 ages in the 45–49 interval. The
boundaries of the 45–49 interval are 44.5 to 49.5
median
Median 5 44.5 1
112
10
35
5 44.5 1 0.75 5 45.25 45
22
44.5
1 12
23
24
25
26
49.5
10
Answer mean 5 44, median 5 45
Exercises
Writing About Mathematics
1. Adelaide said that since, in Example 2, there are 10 employees whose ages are in the 45–49
interval, there must be two employees of age 45. Do you agree with Adelaide? Explain why
or why not.
2. Gail said that since, in Example 2, there are 10 employees whose ages are in the 45–49 interval, there must at least two employees who are the same age. Do you agree with Gail?
Explain why or why not.
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Statistics
Developing Skills
In 3–8, find the mean, the median, and the mode for each set of data.
3.
6.
4.
xi
fi
5
4
3
2
1
0
6
10
15
11
2
1
xi
fi
10
9
8
7
6
5
4
1
1
3
7
6
2
2
7.
xi
fi
50
40
30
20
10
8
12
17
10
3
5.
xi
fi
$1.10
$1.20
$1.30
$1.40
$1.50
1
5
8
6
6
xi
fi
12
11
10
9
8
7
6
5
7
15
13
16
14
15
9
2
xi
fi
95
90
85
80
75
70
65
60
2
8
12
10
9
3
0
1
8.
9. Find the percentile rank of 2 for the data in Exercise 3.
10. Find the percentile rank of 20 for the data in Exercise 4.
11. Find the percentile rank of 8 for the data in Exercise 5.
12. Find the percentile rank of 6 for the data in Exercise 6.
In 13–18, find the mean and the median for each set of data to the nearest tenth.
13.
xi
fi
21–25
16–20
11–15
6–10
1–5
2
3
12
6
1
14.
xi
fi
91–100 5
81–90
8
71–80 10
61–70
6
51–60
0
41–50
1
15.
xi
fi
$1.51–$1.60
$1.41–$1.50
$1.31–$1.40
$1.21–$1.30
$1.11–$1.20
$1.01–$1.10
2
5
14
4
2
3
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Measures of Central Tendency for Grouped Data
16.
xi
fi
17–19
14–16
11–13
8–10
5–7
20
27
32
39
32
17.
18.
xi
fi
$60–$69
$50–$59
$40–$49
$30–$39
$20–$29
$10–$19
16
5
16
2
5
37
613
xi
fi
0.151–0.160
0.141–0.150
0.131–0.140
0.121–0.130
0.111–0.120
16
5
0
6
0
Applying Skills
19. The table shows the number of correct answers on a test consisting of 15 questions. Find the
mean, the median, and the mode for the number of correct answers.
Correct
Answers
6
7
8
9
10
11
12
13
14
15
Frequency
1
0
1
3
5
8
9
6
5
2
20. The ages of students in a calculus class at a high school
are shown in the table. Find the mean and median age.
Age
Frequency
19
18
17
16
15
2
8
9
1
1
21. Each time Mrs. Taggart fills the tank of her car, she estimates, from the number of miles
driven and the number of gallons of gasoline needed to fill the tank, the fuel efficiency of
her car, that is, the number of miles per gallon. The table shows the result of the last 20
times that she filled the car.
a. Find the mean and the median fuel efficiency (miles per gallon) for her car.
b. Find the percentile rank of 34 miles per gallon.
Miles per
Gallon
32
33
34
35
36
37
38
39
40
Frequency
1
3
2
5
3
3
2
0
1
22. The table shows the initial weights of people enrolled in a weight-loss program. Find the
mean and median weight.
Weight
Frequency
Weight
Frequency
191–200
201–210
211–220
221–230
231–240
241–250
1
1
5
7
10
12
251–260
261–270
271–280
281–290
291–300
13
16
8
5
2
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Statistics
23. In order to improve customer relations, an auto-insurance company surveyed 100 people to
determine the length of time needed to complete a report form following an auto accident.
The result of the survey is summarized in the following table showing the number of minutes needed to complete the form. Find the mean and median amount of time needed to
complete the form.
Minutes
Frequency
26–30
31–35
36–40
41–45
46–50
51–55
56–60
61–65
66–70
2
8
12
15
10
24
26
1
2
Hands-On Activity
Organize the data from the survey in the Hands-On Activity of Section 15-1 using intervals of five
seconds. Use the table to find the mean number of seconds. Compare this result with the mean
found using the individual data values.
15-4 MEASURES OF DISPERSION
The mean and the median of a set of data help us to describe a set of data.
However, the mean and the median do not always give us enough information
to draw meaningful conclusions about the data. For example, consider the following sets of data.
Ages of students on a
middle-school basketball team
Ages of students in a community
center tutoring program
11 12 12 13 13 13
13 13 14 14 14 14
6 8 9 9 10 13
13 15 17 18 19 19
The mean of both sets of data is 13 and the median of both sets of data is
13, but the two sets of data are quite different. We need a measure that indicates
how the individual data values are scattered or spread on either side of the
mean. A number that indicates the variation of the data values about the mean
is called a measure of dispersion.
Range
The simplest of the measures of dispersion is called the range. The range is the
difference between the highest value and the lowest value of a set of data. In the
sets of data given above, the range of ages of students on the middle-school basketball team is 14 2 11 or 3 and the range of the ages of the students in the community center tutoring program is 19 2 6 or 13. The difference in the ranges
indicates that the ages of the students on the basketball team are more closely
grouped about the mean than the ages of the students in the tutoring program.
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Measures of Dispersion
615
The range is dependent on only the largest data value and the smallest data
value. Therefore, the range can be very misleading. For example consider the
following sets of data:
Ages of members of the chess club: 11, 11, 11, 11, 15, 19, 19, 19, 19
Ages of the members of the math club: 11, 12, 13, 14, 15, 16, 17, 18, 19
For each of these sets of data, the mean is 15, the median is 15, and the range
is 8. But the sets of data are very different. The range often does not tell us critical information about a set of data.
Interquartile Range
Another measure of dispersion depends on the first and third quartiles of a distribution. The difference between the first and third quartile values is the
interquartile range. The interquartile range tells us the range of at least 50% of
the data. The largest and smallest values of a set of data are often not representative of the rest of the data. The interquartile range better represents the
spread of the data. It also gives us a measure for identifying extreme data values, that is, those that differ significantly from the rest of the data. For example,
consider the ages of the members of a book club.
21, 24, 25, 27, 28, 31, 35, 35, 37, 39, 40, 41, 69
↑
↑
↑
26
35
39.5
median
Q3
Q1
For these 13 data values, the median is the 7th value. For the six values
below the median, the first quartile is the average of the 3rd and 4th values from
27
the lower end: 25 1
5 26. The third quartile is the average of the 3rd and
2
40
4th values from the upper end: 39 1
5 39.5. The interquartile range of
2
the ages of the members of the book club is 39.5 2 26 or 13.5. The age of the
oldest member of the club differs significantly from the ages of the others. It is
more than 1.5 times the interquartile range above the upper quartile:
69 . 39.5 1 1.5(13.5)
We call this data value an outlier.
DEFINITION
An outlier is a data value that is greater than the upper quartile plus 1.5 times
the interquartile range or less than the lower quartile minus 1.5 times the
interquartile range.
When we draw the box-and-whisker plot for a set of data, the outlier is indicated by a ✸ and the whisker is drawn to the largest or smallest data value that
is not an outlier. The box-and-whisker plot for the ages of the members of the
book club is shown on page 616.
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Statistics
*
21
26
39.5 40
35
69
We can use the graphing calculator to graph a box-and-whisker plot with
outliers. From the STAT PLOT menu, choose the ı-▫-ı -- option. For example,
with the book club data entered into L1, the following keystrokes will graph a
box-and-whisker plot with outliers.
ENTER:
2nd
STAT PLOT
1
䉴
ENTER
䉲
䉴
ENTER
䉲
2nd
䉲
ALPHA
DISPLAY:
䉴
L1
1 ZOOM
9
EXAMPLE 1
The table shows the number of minutes, rounded to the nearest 5 minutes,
needed for each of 100 people to complete a survey.
a. Find the range and the interquartile range for this set of data.
b. Does this data set include outliers?
Minutes
30
35
40
45
50
55
60
65
70
85
Frequency
3
8
12
15
10
24
17
8
2
1
Cumulative
Frequency
3
11
23
38
48
72
89
97
99
100
Solution a. The range is the difference between the largest and smallest data value.
Range 5 85 2 30 5 55
To find the interquartile range, we must first find the median and the lower
and upper quartiles. Since there are 100 values, the median is the average of
the 50th and the 51st values. Both of these values lie in the interval 55.
Therefore, the median is 55. There are 50 values above the mean and 50 values below the mean. Of the lower 50 values, the lower quartile is the average of the 25th and 26th values. Both of these values lie in the interval 45.
The lower quartile is 45. The upper quartile is the average of the 25th and
the 26th from the upper end of the distribution (or the 75th and 76th from
the lower end). These values lie in the interval 60. The upper quartile is 60.
Interquartile range 5 60 2 45 5 15
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Measures of Dispersion
617
b. An outlier is a data value that is 1.5 times the interquartile range below the
first quartile or above the third quartile.
45 2 1.5(15) 5 22.5
60 1 1.5(15) 5 82.5
The data value 85 is an outlier.
Answers a. Range 5 55, interquartile range 5 15 b. The data value 85 is an outlier.
Exercises
Writing About Mathematics
1. In any set of data, is it always true that xi 5 i? For example, in a set of data with more than
three data values, does x4 5 4? Justify your answer.
2. In a set of data, Q1 5 12 and Q3 5 18. Is a data value equal to 2 an outlier? Explain why or
why not.
Developing Skills
In 3–6, find the range and the interquartile range for each set of data.
3. 3, 5, 7, 9, 11, 13, 15, 17, 19
4. 12, 12, 14, 14, 16, 18, 20, 22, 28, 34
5. 12, 17, 23, 31, 46, 54, 67, 76, 81, 93
6. 2, 14, 33, 34, 34, 34, 35, 36, 37, 37, 38, 40, 42
In 7–9, find the mean, median, range, and interquartile range for each set of data to the nearest
tenth.
7.
xi
fi
50
45
40
35
30
25
20
3
8
12
15
11
7
4
8.
xi
fi
10
9
8
7
6
5
4
3
2
2
4
6
9
3
3
2
0
1
9.
xi
fi
11
16
19
31
37
32
35
5
8
9
6
5
5
6
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Applying Skills
10. The following data represents the yearly salaries, in thousands of dollars, of 10 basketball
players.
533 427 800 687 264 264 125 602 249 19,014
a. Find the mean and median salaries of the 10 players.
b. Which measure of central tendency is more representative of the data? Explain.
c. Find the outlier for the set of data.
d. Remove the outlier from the set of data and recalculate the mean and median salaries.
e. After removing the outlier from the set of data, is the mean more or less representative of the data?
11. The grades on a math test are shown in the stem-and-leaf
diagram to the right.
Stem
9
8
7
6
5
4
a. Find the mean grade.
b. Find the median grade.
c. Find the first and third quartiles.
d. Find the range.
Leaf
001269
0235557
88
46679
067
68
e. Find the interquartile range.
12. The ages of students in a Spanish class are shown in the table.
Find the range and the interquartile range.
Age
Frequency
19
18
17
16
15
1
8
8
6
2
13. The table shows the number of hours that 40 third graders reported studying a week. Find
the range and the interquartile range.
Hours
3
4
5
6
7
8
9
10
11
12
Frequency
2
1
3
3
5
8
8
5
4
1
14. The table shows the number of pounds lost during the first month by people enrolled in a
weight-loss program.
a. Find the range.
b. Find the interquartile range.
c. Which of the data values is an outlier?
Pounds Lost
1
2
3
4
5
6
7
8
9
11
15
Frequency
1
1
2
2
6
10
7
7
2
1
1
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Variance and Standard Deviation
619
15. The 14 students on the track team recorded the following number of seconds as their best
time for the 100-yard dash:
13.5 13.7 13.1 13.0 13.3 13.2 13.0
12.8 13.4 13.3 13.1 12.7 13.2 13.5
Find the range and the interquartile range.
16. The following data represent the waiting times, in minutes, at Post Office A and Post Office
B at noon for a period of several days.
A: 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 9, 10
B: 1, 2, 2, 3, 3, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10
a. Find the range of each set of data. Are the ranges the same?
b. Graph the box-and-whisker plot of each set of data.
c. Find the interquartile range of each set of data.
d. If the data values are representative of the waiting times at each post office, which post
office should you go to at noon if you are in a hurry? Explain.
Hands-On Activity
Find the range and the interquartile range of the data from the survey in the Hands-On Activity of
Section 15-1 estimating the length of a minute. Does your data contain an outlier?
15-5 VARIANCE AND STANDARD DEVIATION
Variance
Let us consider a more significant measure of dispersion than either the
range or the interquartile range. Let xi represent a student’s grades on
eight tests.
x i 2 x–
(xi 2 x– ) 2
Grade (xi)
95
92
88
87
86
82
80
78
8
9
6
2
1
0
24
26
28
8
81
36
4
1
0
16
36
64
8
a xi 5 688
– 2
a (xi 2 x ) 5 0
– 2
a (xi 2 x ) 5 238
i51
i51
i51
8
a xii
Mean 5
i51
8
5 688
8 5 86
The table shows the deviation,
or difference, of each grade from
the mean. Grades above the mean
are positive and grades below the
mean are negative. For any set of
data, the sum of these differences is
always 0.
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Statistics
In order to find a meaningful sum, we can use the squares of the differences
so that each value will be positive. The square of the deviation of each data value
from the mean is used to find another measure of dispersion called the variance.
To find the variance, find the sum of the squares of the deviations from the mean
and divide that sum by the number of data values. In symbols, the variance for a
set of data that represents the entire population is given by the formula:
n
Variance 5 n1 a (xi 2 x) 2
i51
For the set of data given above, the variance is 18 (238) or 29.75.
Note that since the square of the differences is involved, this method of finding a measure of dispersion gives greater weight to measures that are farther
from the mean.
EXAMPLE 1
A student received the following grades on five math tests: 84, 97, 92, 88, 79. Find
the variance for the set of grades of the five tests.
Solution The mean of this set of grades is:
84 1 97 1 92 1 88 1 79
5
5 440
5 5 88
xi
x i 2 x–
(xi 2 x– ) 2
84
97
92
88
79
24
9
4
0
29
16
81
16
0
81
5
5
a xi 5 440
– 2
a (xi 2 x ) 5 194
i51
i51
– 2 5 A 1 B 194 5 384 5 38.8 Answer
Variance 5 n1 a (xi 2 x)
5
5
5
i51
When the data representing a population is listed in a frequency distribution table, we can use the following formula to find the variance:
n
2
a fi (xi 2 x)
Variance 5
i51
n
a fi
i51
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Variance and Standard Deviation
621
EXAMPLE 2
In a city, there are 50 math teachers who are under the age of 30. The table
below shows the number of years of experience of these teachers. Find the variance of this set of data. Let xi represent the number of years of experience and
fi represent the frequency for that number of years.
xi
0
1
2
3
4
5
6
7
8
fi
1
2
7
8
6
9
8
4
5
50
xifi
0
2
14
24
24
45
48
28
40
225
Solution
8
a xifi
Mean 5
i50
8
a fi
5 225
50 5 4.5
i50
For this set of data, the data value is equal to i for each xi.
The table below shows the deviation from the mean, the square of the deviation from the mean, and the square of the deviation from the mean multiplied
by the frequency.
xi
fi
(xi 2 x– )
(xi 2 x– ) 2
fi(xi 2 x– ) 2
8
7
6
5
4
3
2
1
0
5
4
8
9
6
8
7
2
1
3.5
2.5
1.5
0.5
20.5
21.5
22.5
23.5
24.5
12.25
6.25
2.25
0.25
0.25
2.25
6.25
12.25
20.25
61.25
25.00
18.00
2.25
1.50
18.00
43.75
24.50
20.25
8
8
a fi 5 50
– 2
a fi (xi 2 x ) 5 214.50
i51
i50
8
a
Variance 5
fi (xi 2 x–) 2
i50
8
a fi
5 214.50
50 5 4.29 Answer
i50
Note that the data given in this example represents a population, that is, data for
all of the teachers under consideration.
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Statistics
Standard Deviation Based on the Population
Although the variance is a useful measure of dispersion, it is in square units. For
example, if the data were a set of measures in centimeters, the variance would
be in square centimeters. In order to have a measure that is in the same unit of
measure as the given data, we find the square root of the variance. The square
root of the variance is called the standard deviation. When the data represents
a population, that is, all members of the group being studied:
n
Standard deviation based on a population 5
É
1
n a fi (xi
2 x) 2
i51
If the data is grouped in terms of the frequency of a given value:
n
2
a fi (xi 2 x)
i51
Standard deviation based on a population 5
é
n
a
fi
i51
The symbol for the standard deviation for a set of data that represents a
population is s (lowercase Greek sigma). Many calculators use the symbol sx.
EXAMPLE 3
Find the standard deviation for the number of years of experience for the 50
teachers given in Example 2.
Solution The standard deviation is the square root of the variance.
Therefore:
Standard deviation 5 !4.29 5 2.071231518
Calculator (1) Clear lists 1 and 2.
Solution (2) Enter the number of years of experience in L and the frequency in L .
1
2
(3) Locate the standard deviation for a population (sx) under 1-Var Stats.
ENTER:
STAT
䉴
2nd
L1
2nd
L2
ENTER
DISPLAY:
,
ENTER
>
1–Var Stats
–
x=4.5
x=225
x2=1227
Sx=2.092259788
x=2.071231518
n=50
(4) sx 5 2.071231518
Answer The standard deviation is approximately 2.07.
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Variance and Standard Deviation
623
Standard Deviation Based on a Sample
When the given data is information obtained from a sample of the population,
the formula for standard deviation is obtained by dividing the sum of the squares
of the deviation from the mean by 1 less than the number of data values.
If each data value is listed separately:
n
Standard deviation for a sample 5
É
1
n 2 1 a (xi
2 x–) 2
i51
If the data is grouped in terms of the frequency of a given value:
n
2
a fi (xi 2 x)
Standard deviation for a sample 5
i51
é
n
a a fi b 2 1
i51
The symbol for the standard deviation for a set of data that represents a
sample is s. Many calculators use the symbol sx.
EXAMPLE 4
From a high school, ten students are chosen at random to report their number
of online friends. The data is as follows: 15, 13, 12, 10, 9, 7, 5, 4, 3, and 2.
Solution The total number of online friends for these 10 students is 80 or a mean of 8
online friends (x– 5 8).
Online Friends (xi)
x i 2 x–
(xi 2 x– ) 2
15
13
12
10
9
7
5
4
3
2
7
5
4
2
1
21
23
24
25
26
49
25
16
4
1
1
9
16
25
36
10
– 2
a (xi 2 x ) 5 182
i51
10
Standard deviation 5
É
1
10 2 1 a (xi
i51
2 x) 2 5 #19 (182) 4.5 Answer
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Statistics
EXAMPLE 5
In each of the following, tell whether the population or sample standard deviation should be used.
a. In a study of the land areas of the states of the United States, the area of
each of the 50 states is used.
b. In a study of the heights of high school students in a school of 1,200 students, the heights of 100 students chosen at random were recorded.
c. In a study of the heights of high school students in the United States, the
heights of 100 students from each of the 50 states were recorded.
Solution a. Use the population standard deviation since every state is included. Answer
b. Use the sample standard deviation since only a portion of the total school
population was included in the study. Answer
c. Use the sample standard deviation since only a portion of the total school
population was included in the study. Answer
EXAMPLE 6
A telephone survey conducted in Monroe County obtained information about
the size of the households. Telephone numbers were selected at random until a
sample of 130 responses were obtained. The frequency chart below shows the
result of the survey.
No. of People
per Household
Frequency
1
2
3
4
5
6
7
8
9
28
37
45
8
7
3
1
0
1
Solution The table below can be used to find the mean and the standard deviation.
xi
fi
fixi
(xi 2 x– )
(xi 2 x– ) 2
fi(xi 2 x– ) 2
1
2
3
4
5
6
7
8
9
28
37
45
8
7
3
1
0
1
28
74
135
32
35
18
7
0
9
21.6
20.6
0.4
1.4
2.4
3.4
4.4
5.4
6.4
2.56
0.36
0.16
1.96
5.76
11.56
19.36
29.16
40.96
71.68
13.32
7.20
15.68
40.32
34.68
19.36
0
40.96
9
9
9
a fi 5 130
a fi xi 5 338
– 2
a fi (xi 2 x ) 5 243.20
i51
i51
i51
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Variance and Standard Deviation
625
9
a fixi
Mean 5
i51
9
a fi
5 338
130 5 2.6
i51
9
a
Standard deviation 5
fi (xi 2 x) 2
i51
9
a fi b 2 1
è ia
51
243.20
243.20
5 #130
2 1 5 # 129 5 1.373051826
Calculator Enter the number of members of the households in L1 and the frequency for
Solution each data value in L2.
ENTER:
䉴
STAT
L1
,
ENTER
2nd
L2
2nd
ENTER
DISPLAY:
>
1–Var Stats
–
x=2.6
x=338
x2=1122
Sx=1.373051826
x=1.367760663
n=130
The standard deviation based on the data from a sample is sx 5 1.373051826
or approximately 1.37. Answer
Exercises
Writing About Mathematics
1. The sets of data for two different statistical studies are identical. The first set of data represents the data for all of the cases being studied and the second represents the data for a
sample of the cases being studied. Which set of data has the larger standard deviation?
Explain your answer.
2. Elaine said that the variance is the square of the standard deviation. Do you agree with
Elaine? Explain why or why not.
Developing Skills
In 3–9, the given values represent data for a population. Find the variance and the standard deviation for each set of data.
3. 9, 9, 10, 11, 5, 10, 12, 9, 10, 12, 6, 11, 11, 11
4. 11, 6, 7, 13, 5, 8, 7, 10, 9, 11, 13, 12, 9, 16, 10
5. 20, 19, 20, 17, 18, 19, 42, 41, 41, 39, 39, 40
6. 20, 101, 48, 25, 63, 31, 20, 50, 16, 14, 245, 9
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Statistics
xi
fi
30
35
40
45
50
55
60
1
7
10
9
11
8
6
8.
xi
fi
2
4
6
8
10
12
14
16
0
1
6
10
13
21
7
1
9.
xi
fi
20
25
30
35
40
45
50
55
60
21
10
1
2
2
4
5
3
7
In 10–16, the given values represent data for a sample. Find the variance and the standard deviation
based on this sample.
10. 6, 4, 9, 11, 4, 3, 22, 3, 7, 10
11. 12.1, 33.3, 45.5, 60.1, 94.2, 22.2
12. 15, 10, 16, 19, 10, 19, 14, 17
13. 1, 3, 5, 22, 30, 45, 50, 55, 60, 70
14.
xi
fi
55
50
45
40
35
30
25
11
15
4
1
14
12
4
15.
xi
fi
33
34
35
36
37
38
39
3
1
4
6
5
11
6
16.
xi
fi
1
2
3
4
5
6
7
3
3
3
3
3
3
3
Applying Skills
17. To commute to the high school in which Mr. Fedora teaches, he can take either the Line A
or the Line B train. Both train stations are the same distance from his house and both stations report that, on average, they run 10 minutes late from the scheduled arrival time.
However, the standard deviation for Line A is 1 minute and the standard deviation for Line
B is 5 minutes. To arrive at approximately the same time on a regular basis, which train line
should Mr. Fedora use? Explain.
Stem
Leaf
18. A hospital conducts a study to determine if nurses need extra
staffing at night. A random sample of 25 nights was used. The
number of calls to the nurses’ station each night is shown in
the stem-and-leaf diagram to the right.
a. Find the variance.
b. Find the standard deviation.
9
8
7
6
5
4
0234466
68
01234
446789
037
19
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Variance and Standard Deviation
19. The ages of all of the students in a science class are
shown in the table. Find the variance and the
standard deviation.
Age
Frequency
18
17
16
15
1
2
9
9
627
20. The table shows the number of correct answers on a test consisting of 15 questions. The
table represents correct answers for a sample of the students who took the test. Find the
standard deviation based on this sample.
Correct
Answers
6
7
8
9
10
11
12
13
14
15
Frequency
2
1
3
3
5
8
8
5
4
1
21. The table shows the number of robberies during a given month in 40 different towns of a
state. Find the standard deviation based on this sample
Robberies
0
1
2
3
4
5
6
7
8
9
10
Frequency
1
1
1
2
2
6
10
7
7
2
1
22. Products often come with registration forms. One of the questions usually found on the registration form is household income. For a given product, the data below represents a random sample of the income (in thousands of dollars) reported on the registration form. Find
the standard deviation based on this sample.
38 40 26 42 39 25 40 40 39 36
46 41 43 47 49 43 39 35 43 37
Hands-On Activity
The people in your survey from the Hands-On Activity of Section 15-1 represent a random sample
of all people. Find the standard deviation based on your sample.
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15-6 NORMAL DISTRIBUTION
The Normal Curve
....
......
Imagine that we were able to determine
......
........
........
..........
the height in centimeters of all 10-year..........
..........
............
............
old children in the United States. With a
.............
..............
..............
...............
scale along the horizontal axis that
................
................
..................
...................
includes all of these heights, we will place
....................
.....................
......................
........................
a dot above each height for each child of
.........................
.
...........................
.
.............................
. ................................
that height. For example, we will place
. ..................................
........................................
..........................................
above 140 on the horizontal scale a dot
for each child who is 140 centimeters tall. 123 128 133 138 143 148 153
Do this for 139, 138, 137, and so on for
Height in centimeters
each height in our data. The result would
be a type of frequency histogram. If we
draw a smooth curve joining the top dot for each height, we will draw a bellshaped curve called the normal curve. As the average height of 10-year-olds is
approximately 138 centimeters, the data values are concentrated at 138 centimeters and the normal curve has a peak at 138 centimeters. Since for each height
that is less than or greater than 138 centimeters there are fewer 10-year-olds, the
normal curve progressively gets shorter as you go farther from the mean.
Scientists have found that large sets of data that occur naturally such as
heights, weights, or shoe sizes have a bell-shaped or a normal curve. The highest
point of the normal curve is at the mean of the data. The normal curve is symmetric with respect to a vertical line through the mean of the distribution.
Standard Deviation and the Normal Curve
A normal distribution is a set of data that can be represented by a normal curve.
For a normal distribution, the following relationships exist.
1. The mean and the median of the
data values lie on the line of symmetry of the curve.
2. Approximately 68% of the data
values lie within one standard
deviation from the mean.
99.7% of data values
95% of data values
68% of
data values
3. Approximately 95% of the data
values lie within two standard
deviations from the mean.
4. Approximately 99.7% of the data
values lie within three standard
deviations from the mean.
13.5% 34% 34% 13.5%
x–23s x–22s x–2s x– x–1s x–12s x–13s
Normal distribution
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Normal Distribution
629
EXAMPLE 1
A set of data is normally distributed with a mean of 50 and a standard deviation
of 2.
a. What percent of the data values are less than 50?
b. What percent of the data values are between 48 and 52?
c. What percent of the data values are between 46 and 54?
d. What percent of the data values are less than or equal to 46?
Solution a. In a normal distribution, 50% of the
data values are to the left and 50% to
the right of the mean.
b. 48 and 52 are each 1 standard deviation away from the mean.
48 5 50 2 2
2.5%
13.5% 34% 34% 13.5%
52 5 50 1 2
46
48
50
50%
Therefore, 68% of the data values are
between 48 and 52.
52
54
50%
c. 46 and 53 are each 2 standard deviations away from the mean.
46 5 50 2 2(2)
54 5 50 1 2(2)
Therefore, 95% of the data values are between 48 and 52.
d. 50% of the data values are less than 50.
47.5% of the data values are more than 46 and less than 50.
Therefore, 50% 2 47.5% or 2.5% of the data values are less than or equal
to 46.
Answers a. 50% b. 68% c. 95% d. 2.5%
Z-Scores
The z-score for a data value is the deviation from the mean divided by the standard deviation. Let x be a data value of a normal distribution.
x2x
x2x
z-score 5 standard
deviation 5 s
The z-score of x, a value from a normal distribution, is positive when x is
above the mean and negative when x is below the mean. The z-score tells us how
many standard deviations x is above or below the mean.
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Statistics
1. The z-score of the mean is 0.
99.7%
95%
2. Of the data values, 34% have a
z-score between 0 and 1 and 34%
have a z-score between 21 and 0.
Therefore, 68% have a z-score
between 21 and 1.
68%
3. Of the data values, 13.5% have a
z-score between 1 and 2 and 13.5%
have a z-score between 22 and 21.
23
13.5% 34% 34% 13.5%
22 21
0
1
2
3
4. Of the data values, (34 1 13.5)% or
47.5% have a z-score between 0 and
2 and 47.5% have a z-score between 22 and 0. Therefore, 95% have a
z-score between 22 and 2.
5. Of the data values, 99.7% have a z-score between 23 and 3.
For example, the mean height of 10-year-old children is 138 centimeters
with a standard deviation of 5. Casey is 143 centimeters tall.
138
z-score for Casey 5 143 2
5 55 5 1
5
Casey’s height is 1 standard deviation above the mean. For a normal distribution, 34% of the data is between the mean and 1 standard deviation above the
mean and 50% of the data is below the mean. Therefore, Casey is as tall as or
taller than (34 1 50)% or 84% of 10-year-old children.
A calculator will give us this same answer.
The second entry of the DISTR menu is
normalcdf(. When we use this function, we must
supply a minimum value, a maximum value, the
mean, and the standard deviation separated by
commas:
>
DISTR DRAW
1: normalpdf(
2:normalcdf(
3:invNorm(
4:invT(
5:tpdf(
6:tcdf(
7 x2 p d f (
normalcdf(minimum, maximum, mean, standard deviation)
For the minimum value we can use 0. To find the proportion of 10-year-old
children whose height is 143 centimeters or less, use the following entries.
ENTER:
2nd
143
,
5
DISTR
)
,
ENTER
0
2
138
,
DISPLAY:
normalcdf(0,143,
138,5)
.8413447404
The calculator returns the number 0.8413447404, which can be rounded to 0.84
or 84%.
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Normal Distribution
631
If we wanted to find the proportion of the 10-year-old children who are
between 134.6 and 141.4 centimeters tall, we could make the following entry:
ENTER:
2nd
141.4
,
5
DISTR
)
,
2
138
134.6
DISPLAY:
,
ENTER
normalcdf(134.6,
141.4,138,5)
.5034956838
The calculator returns the number 0.5034956838, which can be rounded to 0.50
or 50%. Since 134.6 and 141.4 are equidistant from the mean, 25% of the data
is below the mean and 25% is above the mean. Therefore, for this distribution,
134.6 centimeters is the first quartile, 138 centimeters is the median or second
quartile, and 141.4 centimeters is the third quartile.
Note: 134.6 is 0.68 standard deviation
below the mean and 141.4 is 0.68 standard
deviation above the mean. For any normal
distribution, data values with z-scores of
20.68 are approximately equal to the first
quartile and data values with z-scores of
0.68 are approximately equal to the third
quartile.
25%
25%
20.68
25%
0
25%
0.68
EXAMPLE 1
On a standardized test, the test scores are normally distributed with a mean of
60 and a standard deviation of 6.
a. Of the data, 84% of the scores are at or below what score?
b. Of the data, 16% of the scores are at or below what score?
c. What is the z-score of a score of 48?
d. If 2,000 students took the test, how many would be expected to score at or
below 48?
Solution a. Since 50% scored at or below the mean and 34% scored within 1 standard
deviation above the mean, (50 1 34)% or 84% scored at or below 1 deviation above the mean:
x– 1 s 5 60 1 6 5 66
b. Since 50% scored at or below the mean and 34% scored within 1 standard
deviation below the mean, (50 2 34)% or 16% scored at or below 1 deviation below the mean:
x– 1 s 5 60 2 6 5 54
–
x
48 2 60
c. z-score 5 x 2
5 212
s 5
6
6 5 22
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Statistics
d. A test score of 48 has a z-score of 22. Since 47.5% of the scores are
between 22 and 0 and 50% of the scores are less than 0, 50% 2 47.5%
or 2.5% scored at or below 48.
2.5% 3 2,000 5 0.025 3 2,000 5 50 students
Answers a. 66 b. 54 c. 22 d. 50
EXAMPLE 2
For a normal distribution of weights, the mean weight is 160 pounds and a
weight of 186 pounds has a z-score of 2.
a. What is the standard deviation of the set of data?
b. What percent of the weights are between 155 and 165?
2 x–
s
186 2 160
s
Solution a. z-score 5 x
25
2s 5 26
s 5 13 Answer
b. ENTER: 2nd
155
13
DISTR
,
)
165
DISPLAY:
2
,
ENTER
160
,
normalcdf(155,16
5,160,13)
.2994775047
About 30% of the weights are between 155 and 165. Answer
Exercises
Writing About Mathematics
1. A student’s scores on five tests were 98, 97, 95, 93, and 67. Explain why this set of scores
does not represent a normal distribution.
2. If 34% of the data for a normal distribution lies between the mean and 1 standard deviation
above the mean, does 17% of the data lie between the mean and one-half standard deviation above the mean? Justify your answer.
Developing Skills
In 3–9, for a normal distribution, determine what percent of the data values are in each given range.
3. Between 1 standard deviation below the mean and 1 standard deviation above the mean
4. Between 1 standard deviation below the mean and 2 standard deviations above the mean
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Normal Distribution
633
5. Between 2 standard deviations below the mean and 1 standard deviation above the mean
6. Above 1 standard deviation below the mean
7. Below 1 standard deviation above the mean
8. Above the mean
9. Below the mean
10. A set of data is normally distributed with a mean of 40 and a standard deviation of 5. Find a
data value that is:
a. 1 standard deviation above the mean
b. 2.4 standard deviations above the mean
c. 1 standard deviation below the mean
d. 2.4 standard deviations below the mean
Applying Skills
In 11–14, select the numeral that precedes the choice that best completes the statement or answers
the question.
11. The playing life of a Euclid mp3 player is normally distributed with a mean of 30,000 hours
and a standard deviation of 500 hours. Matt’s mp3 player lasted for 31,500 hours. His mp3
player lasted longer than what percent of other Euclid mp3 players?
(1) 68%
(2) 95%
(3) 99.7%
(4) more than 99.8%
12. The scores of a test are normally distributed. If the mean is 50 and the standard deviation is
8, then a student who scored 38 had a z-score of
(1) 1.5
(2) 21.5
(3) 12
(4) 212
13. The heights of 10-year-old children are normally distributed with a mean of 138 centimeters
with a standard deviation of 5 centimeters. The height of a 10-year-old child who is as tall as
or taller than 95.6% of all 10-year-old children is
(1) between 138 and 140 cm.
(3) between 145 and 148 cm.
(2) between 140 and 145 cm.
(4) taller than 148 cm.
14. The heights of 200 women are normally distributed. The mean height is 170 centimeters
with a standard deviation of 10 centimeters. What is the best estimate of the number of
women in this group who are between 160 and 170 centimeters tall?
(1) 20
(2) 34
(3) 68
(4) 136
15. When coffee is packed by machine into 16-ounce cans, the amount can vary. The mean
weight is 16.1 ounces and the standard deviation is 0.04 ounce. The weight of the coffee
approximates a normal distribution.
a. What percent of the cans of coffee can be expected to contain less than 16 ounces of
coffee?
b. What percent of the cans of coffee can be expected to contain between 16.0 and 16.2
ounces of coffee?
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Statistics
16. The length of time that it takes Ken to drive to work represents a normal distribution with a
mean of 25 minutes and a standard deviation of 4.5 minutes. If Ken allows 35 minutes to get
to work, what percent of the time can he expect to be late?
17. A librarian estimates that the average number of books checked out by a library patron is 4
with a standard deviation of 2 books. If the number of books checked out each day approximates a normal distribution, what percent of the library patrons checked out fewer than
7 books yesterday?
18. The heights of a group of women are normally distributed with a mean of 170 centimeters
and a standard deviation of 10 centimeters. What is the z-score of a member of the group
who is 165 centimeters tall?
19. The test grades for a standardized test are normally distributed with a mean of 50. A grade
of 60 represents a z-score of 1.25. What is the standard deviation of the data?
20. Nora scored 88 on a math test that had a mean of 80 and a standard deviation of 5. She also
scored 80 on a science test that had a mean of 70 and a standard deviation of 3. On which
test did Nora perform better compared with other students who took the tests?
15-7 BIVARIATE STATISTICS
Statistics are often used to compare two sets of data. For example, a pediatrician
may compare the height and weight of a child in order to monitor growth. Or
the owner of a gift shop may record the number of people who enter the store
with the revenue each day. Each of these sets of data is a pair of numbers and is
an example of bivariate statistics.
Representing bivariate sta550
tistics on a two-dimensional
graph or scatter plot can help
540
us to observe the relationship
530
between the variables. For example the mean value for the critical
520
reading and for the math sections
of the SAT examination for nine
510
schools in Ontario County are
500
listed in the table and shown on
the graph on the right.
510 520 530 540 550 560 570
Critical Reading
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Math
Math
530
551
521
522
537
511
516
537
566
Critical Reading
530
529
512
518
526
500
504
515
543
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Bivariate Statistics
635
The graph shows that there appears to be a linear relationship between the
critical reading scores and the math scores. As the math scores increase, the critical reading scores also increase. We say that there is a correlation between the
two scores. The points of the graph approximate a line.
These data can also be shown on a calculator. Enter the math scores as L1
and the corresponding critical reading scores as L2. Then turn on Plot 1 and use
ZoomStat from the ZOOM menu to construct a window that will include all
values of x and y:
ENTER:
2nd
STAT PLOT
䉲
ENTER
䉲
2nd
DISPLAY:
2nd
ENTER:
1
ZOOM
9
ENTER
L1
䉲
L2
DISPLAY:
Plot1 Plot2 Plot3
On
Off
.
Ty p e : ......
Xlist:L1
Xlist:L2
Mark:
+ .
The calculator will display a graph similar to
that shown above. To draw a line that approximates the data, use the regression line on the calculator. A regression line is a special line of best
fit that minimizes the square of the vertical distances to each data point. In this course, you do
not have to know the formula to find the regression line. The calculator can be used to determine
the regression line:
ENTER:
STAT
䉴
4
VARS
䉴
1
1
The calculator displays values for a and b for
the linear equation y = ax + b and stores the
regression equation into Y1 in the Y menu.
9
Press ZOOM
to display the scatter plot
and the line that best approximates the data. If
we round the given values of a and b to three decimal places, the linear regression equation is:
y 5 0.693x 1 151.013
LinReg
y=ax+b
a=.6925241158
b=151.0129957
r2=.7986740639
r=.8936856628
ENTER
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Statistics
We will study other data that can be approximated by a curve rather than a line
in later sections.
The scatter plots below show possible linear correlation between elements
of the pairs of bivariate data. The correlation is positive when the values of the
second element of the pairs tend to increase when the values of the first elements of the pairs increase. The correlation is negative when the values of the
second element of the pairs tend to decrease when the values of the first elements of the pairs increase.
Strong positive correlation
Moderate positive correlation
No linear correlation
Moderate negative correlation
Strong negative correlation
EXAMPLE 1
Jacob joined an exercise program to try to lose weight. Each month he records
the number of months in the program and his weight at the end of that month.
His record for the first twelve months is shown below:
Month
Weight
1
2
3
4
5
6
7
248 242 237 228 222 216 213
8
9
206
197
10
11
12
193 185 178
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Bivariate Statistics
637
a. Draw a scatter plot and describe the correlation between the data (if any).
b. Draw a line that appears to represent the data.
c. Write an equation of a line that best represents the data.
Solution a. Use a scale of 0 to 13 along a horizontal line for the number of months in
the program and a scale from 170 to 250 along the vertical axis for his
weight at the end of that month. The scatter plot
250
is shown here. There appears to be a strong neg240
ative correlation between the number of months
230
and Jacob’s weight.
Weight
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220
210
200
190
180
170
b. The line on the graph that appears to approximate the data intersects the points (1, 248)
and (11, 185). We can use these two points to
write an equation of the line.
2 185
y 2 248 5 248
1 2 11 (x 2 1)
y 2 248 5 26.3x 1 6.3
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Month
y 5 26.3x 1 254.3
c. On a calculator, enter the number of the
month in L1 and the weight in L2. To use the
calculator to determine a line that approximates the data, use the following sequence:
䉴
ENTER: STAT
1
䉴
VARS
4
1
LinReg
y=ax+b
a=-6.283216783
b=254.5909091
r2=.9966845178
r=-.9983408825
ENTER
The calculator displays values for a and b for the linear equation y = ax 1 b
and stores the equation as Y1 in the Y menu. If we round the given values of a and b to three decimal places, the linear regression equation is:
y 5 26.283x 1 254.591
Turn on Plot 1, then use ZoomStat to graph
the data:
ENTER:
2nd
䉲
STAT PLOT
ENTER
ENTER
2nd
ZOOM
9
䉲
ENTER
2nd
L2
ENTER
L1
ENTER
The calculator will display the scatter plot of the data along with the
regression equation.
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Statistics
EXAMPLE 2
In order to assist travelers in planning a trip, a travel guide lists the average high
temperature of most major cities. The listing for Albany, New York is given in
the following table.
Month
1
2
3
4
5
6
7
8
9
10
11
12
Temperature
31
33
42
51
70
79
84
81
73
62
48
35
Can these data be represented by a regression line?
Solution Draw a scatter plot the data. The
graph shows that the data can be
characterized by a curve rather
than a line. Finding the regression
line for this data would not be
appropriate.
SUMMARY
• The slope of the regression line gives the direction of the correlation:
A positive slope shows a positive correlation.
A negative slope shows a negative correlation.
• The regression line is appropriate only for data that appears to be linearly
related. Do not calculate a regression line for data with a scatter plot
showing a non-linear relationship.
• The regression equation is sensitive to rounding. Round the coefficients to
at least three decimal places.
Exercises
Writing About Mathematics
1. Explain the difference between univariate and bivariate data and give an example of each.
2. What is the relationship between slope and correlation? Can slope be used to measure the
strength of a correlation? Explain.
Developing Skills
In 3–6, is the set of data to be collected univariate or bivariate?
3. The science and math grades of all students in a school
4. The weights of the 56 first-grade students in a school
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Bivariate Statistics
639
5. The weights and heights of the 56 first-grade students in a school
6. The number of siblings for each student in the first grade
In 7–10, look at the scatter plots and determine if the data sets have high linear correlation, moderate linear correlation, or no linear correlation.
7.
8.
9.
10.
Applying Skills
In 11–17: a. Draw a scatter plot. b. Does the data set show strong positive linear correlation, moderate positive linear correlation, no linear correlation, moderate negative linear correlation, or
strong negative linear correlation? c. If there is strong or moderate correlation, write the equation
of the regression line that approximates the data.
11. The following table shows the number of gallons of gasoline needed to fill the tank of a car
and the number of miles driven since the previous time the tank was filled.
Gallons
8.5
7.6
9.4
8.3 10.5 8.7
9.6
4.3
6.1
7.8
Miles
255 230 295 250 315 260 290 130
180
235
12. A business manager conducted a study to examine the relationship between number of ads
placed for each month and sales. The results are shown below where sales are in the thousands.
Number of Ads
10
12
14
Sales
20 26.5 32
16
18
34.8
40
20
22
24
26
28
30
47.2 49.1 56.9 57.9 65.8 66.4
13. Jack Sheehan looked through some of his favorite recipes to compare the number of calories per serving to the number of grams of fat. The table below shows the results.
Calories
310 210 260 330 290 320 245 293
Fat
11
5
11
12
14
16
7
10
220 260 350
8
8
15
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Statistics
14. Greg did a survey to support his theory that the size of a family is related to the size of the
family in which the mother of the family grew up. He asked 20 randomly selected people to
list the number of their siblings and the number of their mother’s siblings. Greg made the
following table.
Family
2
2
3
1
0
3
5
2
1
2
Mother’s Family
4
0
3
7
4
2
2
6
4
7
Family
3
6
0
2
1
4
1
0
4
1
Mother’s Family
5
4
6
1
0
2
1
3
3
2
15. When Marie bakes, it takes about five and a half minutes for the temperature of the oven to
reach 350°. One day, while waiting for the oven to heat, Marie recorded the temperature
every 20 seconds. Her record is shown below.
Seconds
0
20
40
60
80
100 120
140
160
Temperature
100 114 126 145 160 174 193
207
222
Seconds
180 200 220 240 260 280 300
320
340
Temperature
240 255 268 287 301 318 331
342
350
16. An insurance agent is studying the records of his insurance company looking for a relationship between age of a driver and the percentage of accidents due to speeding. The table
shown below summarizes the findings of the insurance agent.
Age
17
18
21
25
30
35
40
45
50
55
60
65
% of Speeding
Accidents
49
49
48
39
31
33
24
25
16
10
5
6
17. A sociologist is interested in the relationship between body weight and performance on the
SAT. A random sample of 10 high school students from across the country provided the following information:
Weight
Score
197
193
194
157
159
170
149
169
157
185
1,485 1,663 1,564 1,300 1,668 1,405 1,544 1,752 1,395 1,214
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Correlation Coefficient
641
15-8 CORRELATION COEFFICIENT
We would like to measure the strength of the linear relationship between the
variables in a set of bivariate data. The slope of the regression equation tells us
the direction of the relationship but it does not tell us the strength of the relationship. The number that we use to measure both the strength and direction of
the linear relationship is called the correlation coefficient, r. The value of the
correlation coefficient does not depend on the units of measurement. In more
advanced statistics courses, you will learn a formula to derive the correlation
coefficient. In this course, we can use the graphing calculator to calculate the
value of r.
EXAMPLE 1
The coach of the basketball team made the following table of attempted and
successful baskets for eight players.
Attempted
Baskets (xi)
10
12
12
13
14
15
17
19
Successful
Baskets (yi)
6
7
9
8
10
11
14
15
Find the value of the correlation coefficient.
Solution Enter the given xi in L1 and yi in L2.
Then choose LinReg(ax+b) from the
CALC STAT menu:
ENTER:
STAT
䉴
4
ENTER
LinReg
y=ax+b
a=1.066666667
b=-4.933333333
r2=.9481481481
r=.9737289911
The calculator will list both the regression equation and r, the correlation coefficient.
Answer r 5 0.97
Note: If the correlation coefficient does not appear on your calculator, enter
2nd
CATALOG
D , scroll down to DiagnosticOn, press ENTER , and
press ENTER again.
When the absolute value of the correlation coefficient is close to 1, the data
have a strong linear correlation. When the absolute value of the correlation
coefficient is close to 0, there is little or no linear correlation. Values between 0
and 1 indicate various degrees of positive moderate correlation and values
between 0 and 21 indicate various degrees of negative moderate correlation.
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Statistics
Properties of the Correlation Coefficient, r
21 r 1.The correlation coefficient is a number between 21 and 1.
When r 5 1 or 21, there is a perfect linear
relationship between the data values.
r 5 1
r 5 21
When r 5 0, no linear relationship exists
between the data values.
When |r| is close to 1, the data have a
strong linear relationship.Values between
0 and 1 indicate various degrees of
moderate correlation.
The sign of r matches the sign of the slope of
the regression line.
EXAMPLE 2
An automotive engineer is studying the fuel efficiency of a new prototype. From
a fleet of eight prototypes, he records the number of miles driven and the number of gallons of gasoline used for each trip.
Miles Driven
310
270
350
275
380
320
290
405
Gallons of
Gasoline Used
10.0
9.0
11.2
8.7
12.3
10.2
9.5
12.7
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Correlation Coefficient
643
a. Based on the context of the problem, do you think the correlation coefficient will be positive, negative, or close to 0?
b. Based on the scatter plot of the data, do you expect the correlation coefficient to be close to 21, 0, or 1?
c. Use a calculator to find the equation of the regression line and determine
the correlation coefficient.
Solution a. As gallons of gasoline used tend to increase with miles driven, we expect
the correlation coefficient to be positive.
Enter the given data as L1 and L2 on a calculator.
b.
c.
There appears to be a strong positive
correlation, so r will be close to 1.
y 5 0.030x 1 0.748, r 5 0.99
LinReg
y=ax+b
a=.0298533724
b=.7476539589
r2=.9879952255
r=.9939794895
EXAMPLE 3
The produce manager of a food store noted the relationship between the
amount the store charged for a pound of fresh broccoli and the number of
pounds sold in one week. His record for 11 weeks is shown in the following
table.
Cost per
Pound
Pounds
Purchased
$0.65 $0.85 $0.90 $1.00 $1.25 $1.50 $1.75 $1.99 $2.25 $2.50 $2.65
58
43
49
23
39
16
56
32
12
35
What conclusion could the product manager draw from this information?
11
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Statistics
Solution Enter the given data as L1 and L2 on the calculator, graph the scatter plot, and
find the value of the correlation coefficient.
LinReg
y=ax+b
a=-13.82881393
b=55.73638117
r2=.331571737
r=-.5758226611
From the scatter plot, we can see that there is a moderate negative correlation.
Since r 5 20.58, the product manager might conclude that a lower price does
explain some of the increase in sales but other factors also influence the number of sales.
A Warning About Cause-and-Effect
The correlation coefficient is a number that measures the strength of the linear
relationship between two data sets. However, simply because there appears to
be a strong linear correlation between two variables does not mean that one
causes the other. There may be other variables that are the cause of the
observed pattern. For example, consider a study on the population growth of a
city. Although a statistician may find a linear pattern over time, this does not
mean that time causes the population to grow. Other factors cause the city grow,
for example, a booming economy.
SUMMARY
• 21 r 1. The correlation coefficient is a number between 21 and 1.
• When r 5 1 or r 5 21, there is a perfect linear relationship between the
data values.
• When r 5 0, no linear relationship exists between the data values.
• When r is close to 1, the data have a strong linear relationship. Values
between 0 and 1 indicate various degrees of moderate correlation.
• The sign of r matches the sign of the slope of the regression line.
• A high correlation coefficient does not necessarily mean that one variable
causes the other.
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Correlation Coefficient
645
Exercises
Writing About Mathematics
1. Does a correlation coefficient of 21 indicate a lower degree of correlation than a correlation coefficient of 0? Explain why or why not.
2. If you keep a record of the temperature in degrees Fahrenheit and in degrees Celsius for a
month, what would you expect the correlation coefficient to be? Justify your answer.
Developing Skills
In 3–6, for each of the given scatter plots, determine whether the correlation coefficient would be
close to 21, 0, or 1.
3.
4.
5.
6.
In 7–14, for each of the given correlation coefficients, describe the linear correlation as strong positive, moderate positive, none, moderate negative, or strong negative.
7. r 5 0.9
11. r 5 1
8. r 5 21
9. r 5 20.1
12. r 5 20.5
10. r 5 0.3
13. r 5 0
14. r 5 20.95
Applying Skills
In 15–19: a. Draw a scatter plot for each data set. b. Based on the scatter plot, would the correlation
coefficient be close to 21, 0, or 1? Explain. c. Use a calculator to find the correlation coefficient for
each set of data.
15. The following table shows the number of gallons of gasoline needed to fill the tank of a car
and the number of miles driven since the previous time the tank was filled.
Gallons 12.5 3.4
Miles
7.9
9.0 15.7 7.0
5.1 11.9 13.0 10.7
392 137 249 308 504 204 182 377
407
304
16. A man on a weight-loss program tracks the number of pounds that he lost over the course
of 10 months. A negative number indicates that he actually gained weight for that month.
Month
No. of
Pounds Lost
1
2
3
4
5
6
7
8
9.2
9.1
4.8
4.5
2.8
1.8
1.2
0
9
10
0.8 22.6
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17. An economist is studying the job market in a large city conducts of survey on the number of
jobs in a given neighborhood and the number of jobs paying $100,000 or more a year. A
sample of 10 randomly selected neighborhood yields the following data:
Total Number
of Jobs
24
28
17
39
32
21
39
39
24
29
No. of HighPaying Jobs
3
3
4
5
7
3
4
7
7
4
18. The table below shows the same-day forecast and the actual high temperature for the day
over the course of 18 days. The temperature is given in degrees Fahrenheit.
Same-Day Forecast
56
52
67
55
58
56
59
57
53
Actual Temperature
53
54
63
49
66
54
54
56
59
Same-Day Forecast
45
55
45
58
59
55
48
53
54
Actual Temperature
48
60
36
59
59
47
46
52
48
19. The table below shows the five-day forecast and the actual high temperature for the fifth
day over the course of 18 days. The temperature is given in degrees Fahrenheit.
Five-Day Forecast
56
52
67
55
58
56
59
57
53
Actual Temperature
50
50
84
54
57
40
70
79
48
Five-Day Forecast
45
55
45
58
59
55
48
53
54
Actual Temperature
40
61
40
70
46
75
49
46
88
20. a. In Exercises 18 and 19, if the forecasts were 100% accurate, what should the value of r
be?
b. Is the value of r for Exercise 18 greater than, equal to, or less than the value of r for
Exercise 19? Is this what you would expect? Explain.
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15-9 NON-LINEAR REGRESSION
Not all bivariate data can be represented
by a linear function. Some data can be
better approximated by a curve. For
example, on the right is a scatter plot of
the file size of a computer program called
Super Type over the course of 6 different
versions. The relationship does not appear
to be linear. For this set of data, a linear
regression would not be appropriate.
There are a variety of non-linear
functions that can be applied to non-linear data. In a statistics course, you will
learn more rigorous methods of determining the regression model. In this
course, we will use the scatter plot of the
data to choose the regression model:
Regression to Use
Description of Scatter Plot
Exponential
• An exponential curve that
does not pass through (0, 0)
• y-intercept is positive
• Data constraint: y . 0
Logarithmic
• A logarithmic curve that
does not pass through (0, 0)
• y-intercept is positive or
negative
• Data constraint: x . 0
Power
• Positive half of power curve
passing through (0, 0)
• Data constraints: x . 0, y . 0
1,800
Size (megabytes)
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1,600
1,400
1,200
1,000
800
600
400
200
0
0
1
2
3 4
5
6
7
Version
Examples
(continued on next page)
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Regression to Use
Description of Scatter Plot
Examples
Specific types of power regression:
Quadratic
• A quadratic curve
Cubic
• A cubic curve
The different non-linear regression models can be found in the STAT
CALC menu of the graphing calculator.
• 5:QuadReg is quadratic regression.
• 6:CubicReg is cubic regression.
• 9:LnReg is logarithmic regression.
• 0:ExpReg is exponential regression.
• A:PwrReg is power regression.
In the example of the file size of Super Type, the scatter plot appears to be
exponential or power. The table below shows the data of the scatter plot:
Version
Size (megabytes)
1
2
3
4
5
6
155
240
387
630
960
1,612
To find the exponential regression model, enter the data into L1 and L2.
Choose ExpReg from the STAT CALC menu:
ENTER:
STAT
䉴
0
VARS
䉴
1
1
ENTER
The calculator will display the regression equation and store the equation into
Y1 of the Y menu. To the nearest thousandth, the regression equation is
y 5 95.699(1.596x). Press ZOOM
sion equation.
9
to graph the scatter plot and the regres-
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ExpReg
y=a* b^x
a=95.69902108
b=1.595666689
r2=.9994598905
r=.9997299088
To find the power regression model, with the data in L1 and L2, choose
PwrReg from the STAT CALC menu:
ENTER:
STAT
䉴
ALPHA
䉴
VARS
A
1
ENTER
1
The calculator will display the regression equation and store the equation into
Y1 of the Y menu. To the nearest thousandth, the regression equation is
y 5 121.591x1.273. Press ZOOM
to graph the scatter plot and the regres-
9
sion equation.
PwrReg
y=a* x^b
a=121.5914377
b=1.273149963
r2=.9307655743
r=.9647619263
From the scatter plots, we see that the exponential regression equation is a
better fit for the data.
EXAMPLE 1
A stone is dropped from a height of 1,000 feet. The trajectory of the stone is
recorded by a high-speed video camera in intervals of half a second. The
recorded distance that the stone has fallen in the first 5 seconds in given below:
Seconds
1
1.5
2
2.5
3
3.5
4
4.5
Distance
16
23
63
105 149 191 260 321
a. Determine which regression model is most appropriate.
b. Find the regression equation. Round all values to the nearest thousandth.
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Solution a. Draw a scatter plot of the data. The data
appears to approximate an exponential
function or a power function. Enter the
data in L1 and L2. Find and graph the exponential and power models. From the
displays, it appears that the power model is
the better fit.
exponential
power
b. To the nearest thousandth, the calculator will display the power equation
y = axb for a 5 13.619, b 5 2.122.
Answers a. Power regression
b. y 5 13.619(x2.122)
EXAMPLE 2
A pediatrician has the following table that lists the head circumferences for a
group of 12 baby girls from the same extended family. The circumference is
given in centimeters.
Age in Months
Circumference
2
2
5
4
1
17
11
14
7
11
10
19
36.8 37.2 38.6 38.2 35.9 40.4 39.7 39.9 39.2 41.1 39.3 40.5
a. Make a scatter plot of the data.
b. Choose what appears to be the curve that best fits the data.
c. Find the regression equation for this model.
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Solution a.
b. The data appears to approximate a log function. Answer
c. With the ages in L1 and the circumferences in
L2, choose LnReg from the STAT CALC
menu:
ENTER:
STAT
䉴
9
ENTER
` The equation of the regression equation, to
the nearest thousandth, is
LnReg
y=a+blnx
a=35.9381563
b=1.627161495
r2=.9281619959
r=.9634116441
y 5 35.938 1 1.627 ln x Answer
Exercises
Writing About Mathematics
1. At birth, the average circumference of a child’s head is 35 centimeters. If the pair (0, 35) is
added to the data in Example 2, the calculator returns an error message. Explain why.
2. Explain when the power function, y = axb, has only positive or only negative y-values and
when it has both positive and negative y-values.
Developing Skills
In 3–8, determine the regression model that appears to be appropriate for the data.
3.
4.
5.
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6.
7.
8.
In 9–13: a. Create a scatter plot for the data. b. Determine which regression model is the most appropriate for the data. Justify your answer. c. Find the regression equation. Round the coefficient of the
regression equation to three decimal places.
9.
10.
11.
12.
x
4
7
3
8
6
5
6
3
9
4.5
y
10
7
15
9
5
6
6
14
14
8
x
–2.3 1.8 21.0 4.6
1.4
3.7
5.3
1.9
0.7
4.2
y
2.2 18.3 8.2
x
3.3
23.8 22.1
0.4
3.5
23.8 21.8 20.4
2.4
1.2
y
12.5 217.1 23.6
0.4
15.0 218.9 22.1 20.4
4.2
3.4
x
y
13.
1
2
63.7 15.3 43.0 89.5 22.7 12.1 54.7
3
4
5
6
7
8
9
10
27 25.6 24.8 24.2 23.8 23.4 23.1 22.8 22.6 22.4
x
1
2
3
4
5
6
7
8
9
10
y
2.7
2.3
2.0
1.7
1.5
1.2
1.1
0.9
0.8
0.7
Applying Skills
14. Mrs. Vroman bought $1,000 worth of shares in the Acme Growth Company. The table below
shows the value of the investment over 10 years.
Year
Value ($)
1
2
3
4
1,045 1,092 1,141 1,192
5
6
7
8
9
10
1,246
1,302
1,361
1,422
1,486
1,553
a. Find the exponential regression equation for the data with the coefficient and base
rounded to three decimal places.
b. Predict, to the nearest dollar, the value of the Vromans’ investment after 11 years.
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15. The growth chart below shows the average height in inches of a group of 100 children from
2 months to 36 months.
Month
2
Height in Inches
4
8
10
12
14
16
18
22.7 26.1 27.5 28.9 31.7 32.1 32.7 33.1 34.0
Month
20
Height in inches
6
22
24
26
28
30
32
34
36
34.4 34.6 34.9 35.2 35.6 36.0 36.6 37.2 37.6
a. Find the logarithmic regression equation for the data with the coefficients rounded to
three decimal places.
b. Predict, to the nearest tenth of an inch, the average height of a child at 38 months.
16. The orbital speed in kilometers per second and the distance from the sun in millions of kilometers of each of six planets is given in the table.
Planet
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Orbital Speed
34.8
29.6
23.9
12.9
9.6
6.6
Distance from the Sun
108.2
149.6
227.9
778.0
1,427
2,871
a. Find the regression equation that appears to be the best fit for the data with the coefficient rounded to three decimal places.
b. Neptune has an orbital speed of 5.45 km/sec and is 4,504 million kilometers from the
sun. Does the equation found for the six planets given in the table fit the data for
Neptune?
17. A mail order company has shipping boxes that have square bases and varying heights
from 1 to 5 feet. The relationship between the height of the box and the volume is shown
in the table.
Height (ft)
3
Volume (ft )
1
1.5
2
2.5
3
3.5
2
7
16
31
54
86
4
4.5
128 182
5
250
a. Create a scatter plot for the data. Let the horizontal axis represent the height of the
box and the vertical axis represent the volume.
b. Determine which regression model is most appropriate for the data. Justify your
answer.
c. Find the regression equation. Round the coefficient of the regression equation to three
decimal places.
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18. In an office building the thermostats have six settings. The table below shows the average
temperature in degrees Fahrenheit for a month that each setting produced.
Setting
1
2
3
4
5
6
Temperature (°F)
61
64
66
67
69
70
a. Create a scatter plot for the data. Let the horizontal axis represent the setting and the
vertical axis represent temperature.
b. Find the equation of best fit using a power regression. Round the coefficient of the
regression equation to three decimal places.
19. The following table shows the speed in megahertz of Intel computer chips over the course
of 36 years. The time is given as the number of years since 1971.
Year
0
1
3
7
11
14
18
22
Speed 0.108
0.8
2
5
6
16
25
66
Year
24
26
28
29
31
34
35
36
Speed
200
300
500
1,500 1,700 3,200 2,900 3,000
One application of Moore’s Law is that the speed of a computer processor should double
approximately every two years. Use this information to determine the regression model.
Does Moore’s Law hold for Intel computer chips? Explain.
Hands-On Activity: Sine Regression
If we make a scatter plot of the following set of data on a graphing calculator, we may observe that
the data points appear to form a sine curve.
x
13
14
15
16
17
y 211.6 217.2 218.0 215.0 28.4
18
19
20
21
22
23
24
6.2
12.0
20.1
18.4
11.9
3.4
26.9
A sine function should be used to model the data. We
can use a graphing calculator to find the sinusoidal regression equation. Enter the x-values into L1 and the y-values
into L2. Then with the calculator in radian mode, choose
SinReg from the STAT CALC menu:
ENTER:
䉴
STAT
1
1
ALPHA
ENTER
C
VARS
䉴
25
20
15
10
5
0
25
210
215
220
10 12 14 16 18 20 22 24 26
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655
To the nearest thousandth, the sinusoidal regression equation is:
y 5 19.251 sin (0.551x 1 2.871) 2 0.029
Press ZOOM
9
to graph the scatter plot and the regression equation.
Note: The sinusoidal regression model on the graphing calculator assumes that the x-values are
equally spaced and in increasing order. For arbitrary data, you need to give the calculator an estimate of the period. See your calculator manual for details.
The average high temperature of a city is recorded for 14 months. The table below shows this
data.
Month
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Temp (°F)
40
48
61 71
81
85
83
77
67
54
41
39
42
49
a. Create a scatter plot for the data.
b. Find the sinusoidal regression equation for the data with the coefficient and base rounded to
three decimal places.
c. Predict the average temperature of the city at 15 months. Round to the nearest degree.
d. Predict the average temperature of the city at 16 months. Round to the nearest degree.
15-10 INTERPOLATION AND EXTRAPOLATION
Data are usually found for specific values of one of the variables. Often we wish
to approximate values not included in the data.
Interpolation
The process of finding a function value between given values is called
interpolation.
EXAMPLE 1
Each time Jen fills the tank of her car, she records the number of gallons of gas
needed to fill the tank and the number of miles driven since the last time that
she filled the tank. Her record is shown in the table.
Gallons of Gas
7.5
8.8
5.3
9.0
8.1
4.7
6.9
8.3
Miles
240 280 170 290 260 150 220
270
a. If Jen needs 8.0 gallons the next time she fills the tank, to the nearest mile,
how many miles will she have driven?
b. If Jen has driven 200 miles, to the nearest tenth, how many gallons of gasoline can she expect to need?
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Solution Graph the scatter plot of the data.
There appears to be positive linear
correlation.
With the data in L1 and L2, use the
calculator to find the regression
equation:
ENTER:
STAT
䉴
4
ENTER
When rounded to the nearest thousandth, the linear equation returned
is
LinReg
y=ax+b
a=32.36537919
b=-2.076402594
r2=.9988025622
r=.9994011018
y 5 32.365x 2 2.076
a. Substitute 8.0 for x in the equation given by the calculator.
y 5 32.365x 2 2.076
5 32.365(8.0) 2 2.076
5 256.844
Jen will have driven approximately 257 miles. Answer
b. Substitute 200 for y in the equation given by the calculator.
y 5 32.365x 2 2.076
200 5 32.365x 2 2.076
202.076 5 32.365x
6.244 x
Jen will need approximately 6.2 gallons of gasoline. Answer
Extrapolation
Often we want to use data collected about past events to predict the future. The
process of using pairs of values within a given range to approximate values outside of the given range of values is called extrapolation.
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EXAMPLE 2
The following table shows the number of high school graduates in the U.S. in the
thousands from 1992 to 2004.
Year
1992 1993 1994
1995 1996 1997 1998
No. of Graduates 2,478 2,481 2,464 2,519 2,518 2,612 2,704
Year
1999 2000 2001
2002 2003 2004
No. of Graduates 2,759 2,833 2,848 2,906 3,016 3,081
a. Write a linear regression equation for this data.
b. If the number of high school graduates continued to grow at this rate, how
many graduates would there have been in 2006?
c. If the number of high school graduates continues to grow at this rate,
when is the number of high school graduates expected to exceed
3.5 million?
Solution a. Enter the year using the number of years
since 1990, that is, the difference between
the year and 1990, in L1. Enter the corresponding number of high school graduates
in L2. The regression equation is:
y 5 53.984x 1 2,277.286 Answer
b. Use the equation y 5 53.984x 1 2,277.286
and let x 5 16:
y 5 53.984(16) 1 2,277.286 3,141
If the increase continued at the same rate, the expected number of graduates in 2006 would have been approximately 3,141,000.
c. Use the equation y 5 53.984x 1 2,277.286 and let y 5 3,500:
3,500 5 53.984x 1 2,277.286
1,222.714 5 53.984x
22.650 x
If the rate of increase continues, the number of high school graduates can
be expected to exceed 3.5 million in the 23rd year after 1990 or in the year
2013.
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Unlike interpolation, extrapolation is not usually accurate. Extrapolation is
valid provided we are sure that the regression model continues to hold outside
of the given range of values. Unfortunately, this is not usually the case. For
instance, consider the data given in Exercise 15 of Section 15-9.
The growth chart below shows the average height in inches of a group of 100
children from 2 months to 36 months.
Month
2
Height in Inches
4
6
8
10
12
14
16
18
22.7 26.1 27.5 28.9 32.1 31.7 33.1 32.7 34.0
Month
20
Height in inches
22
24
26
28
30
32
34
36
34.4 34.6 36.0 34.6 35.2 36.6 35.6 37.2 37.6
The data appears logarithmic. When the
coefficient and the exponent are rounded to
three decimal places, an equation that best fits
the data is y 5 19.165 1 5.026 ln x. If we use this
equation to find the height of child who is 16
years old (192 months), the result is approximately 45.6 inches or less than 4 feet. The average 16-year-old is taller than this. The chart is
intended to give average growth for very young
children and extrapolation beyond the given
range of ages leads to errors.
Exercises
Writing About Mathematics
1. Explain the difference between interpolation and extrapolation.
2. What are the possible sources of error when using extrapolation based on the line of best fit?
Developing Skills
In 3–5: a. Determine the appropriate linear regression model to use based on the scatter plot of the
given data. b. Find an approximate value for y for the given value of x. c. Find an approximate value
for x for the given value of y.
3. b. x 5 5.7 c. y 5 1.25
x
y
1
2
3
4
5
6
7
8
9
10
1.05 1.10 1.16 1.22 1.28 1.34 1.41 1.48 1.55 1.62
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4. b. x 5 12 c. y 5 140
x
1
2
3
4
5
6
7
8
9
10
y
3.1
3.6
4.0
4.5
5.1
5.6
6.0
6.5
6.9
7.5
6
7
8
9
10
5. b. x 5 0.5 c. y 5 0.5
x
1
2
3
4
5
y
1
2.3
3.7
5.3
6.9
8.6 10.3 12.1 14.0 15.9
In 6–9: a. Determine the appropriate non-linear regression model to use based on the scatter plot
of the given data. b. Find an approximate value for y for the given value of x. c. Find an approximate
value for x for the given value of y.
6. b. x 5 1.4 c. y 5 1.50
x
y
1
2
3
4
5
6
7
8
9
10
0.80 1.09 1.25 1.37 1.46 1.54 1.60 1.66 1.71 1.75
7. a. x 5 12 b. y 5 80
x
y
23
26
13
14
11.6 33.3 52.5 43.5
20
17
29
18
18
17
4.0 18.7 84.0 8.0 12.4 11.5
8. a. x 5 10.5 b. y 5 100.0
x
–2.0 21.0 20.5 0.1
0.5
0.8
y
1.0
7.7
9.3 12.0 16.5 21.6 28.0
2.3
3.3
5.3
1.1
1.5
1.8
2.1
9. a. x 5 12 b. y 5 215
x
0.5
1.7
2.7
3.9
4.9
5.7
7.0
8.2
9.2
10.0
y
0.1
2.1
7.8
23.1
43.4
65.1 114.9 183.8 248.3 311.3
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Applying Skills
In 10–12, determine the appropriate linear regression model to use based on the scatter plot of the
given data.
10. The following table represents the percentage of the Gross Domestic Product (GDP) that a
country spent on education.
Year
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Percent
2.71
3.17
5.20
5.61
7.20
8.14
8.79
10.21
10.72
11.77
a. Estimate the percentage of the GDP spent on education in 1998.
b. Assuming this model continues to hold into the future, predict the percentage of the
GDP that will be spent on education in 2015.
11. The following chart gives the average time in seconds that a group of 10 F1 racing cars went
from zero to the given miles per hour.
Speed
75
100 125 150 175 200 225 250 275 300
Time
1.2
2.2
2.9
4.0
5.6
6.8
7.3
8.7
9.3
9.9
a. What was the average time it took the 10 racing cars to reach 180 miles per hour?
b. Estimate the average time it will take the 10 racing cars to reach 325 miles per hour.
12. The relationship between degrees Celsius and degrees Fahrenheit is shown in the table at
intervals of 10° Fahrenheit.
Celsius
0
10
20
30
40
50
60
70
80
90 100
Fahrenheit
32
50
68
86 104 122 140 158 176 194 212
a. Find the Fahrenheit temperature when the Celsius temperature is 25°.
b. Find the Celsius temperature when the Fahrenheit temperature is 24°.
13. The following table gives the number of compact cars produced in a country over the
course of several years.
Year
1981
1984
1987
1990
1993
1996
1999
2002
2005
2008
No. of Cars
100
168
471
603
124
1,780 1,768
4,195
6,680
10,910
a. Estimate the number of cars produced by the country in 2000 using an exponential
model.
b. Estimate the number of cars produced by the country in 1978 using the model from
part a.
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14. In an office building the thermostats have six settings. The table below shows the average
temperature in degrees Fahrenheit for a month that each setting produced.
Setting
1
2
3
4
5
6
Temperature (°F)
61
64
66
67
69
70
Using a power model and assuming that it is possible to choose a setting between the given
settings:
a. what temperature would result from a setting halfway between 2 and 3?
b. where should the setting be placed to produce a temperature of 68 degrees?
In 15 and 16, determine the appropriate non-linear regression model to use based on the scatter plot
of the given data.
15. A mail order company has shipping boxes that have square bases and varying height from
1 to 5 feet. The relationship between the height of the box and the volume in cubic feet is
shown in the table.
Height
1
1.5
2
2.5
3
3.5
4
4.5
5
Volume
2
6.75
16
31.25
54
85.75
128
182.25
250
a. If the company introduces a box with a height of 1.25 feet, what would be the volume
to the nearest hundredth cubic foot?
b. If the company needs a box with a volume of at least 100 cubic feet, what would be the
smallest height to the nearest tenth of a foot?
c. If the company needs a box with a volume of 800 square feet, what would be the
height to the nearest foot?
16. Steve kept a record of the height of a tree that he planted. The heights are shown in the
table.
Age of Tree in Years
1
3
5
Height in Inches
7
12
15
7
9
11
13
16.5 17.8 19
20
a. Write an equation that best fits the data.
b. What was the height of the tree after 2 years?
c. If the height of the tree continues in this same pattern, how tall will the tree be after
20 years?
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CHAPTER SUMMARY
Univariate
Statistics
Statistics is the science that deals with the collection, organization, summarization, and interpretation of related information called data. Univariate statistics consists of one number for each data value. Data can be collected by
means of censuses, surveys, controlled experiments, and observational studies.
The source of information in a statistical survey may be the population, all cases
to which the study applies, or a sample, a representative subset of the entire
group.
The mean, the median, and the mode are the most common measures of central tendency. The mean is the sum of all of the data values divided by the number of data values.
n
a fi xi
Mean 5 i51n
a fi
i51
When finding the mean of data grouped in intervals larger than 1, the
median value of each interval is used to represent each entry in the interval.
The median is the middle number when the data are arranged in numerical
order. In a set of 2n data values arranged in numerical order, the median is the
average of values that are the n and (n 1 1) entries. In a set of 2n 1 1 data values arranged in numerical order, the median is the n 1 1 entry.
The mode is the data value that occurs most frequently.
The quartiles separate the data into four equal parts. The second quartile is
the median. When the data values are arranged in numerical order starting with
the smallest value, the first or lower quartile is the middle value of the values
that precede the median, and the third or upper quartile is the middle value of
the values that follow the median.
An outlier is a data value that is more than 1.5 times the interquartile range
above the third quartile or less than 1.5 times the interquartile range below the
first quartile.
Measures of
Dispersion
• Range 5 the difference between the smallest and largest data values
• Interquartile range 5 Q3 2 Q1
8
a
• Variance based on a population 5
fi (xi 2 x–) 2
i50
8
a fi
i50
n
22
a fi (xi 2 x )
• Standard deviation based on a population 5 s 5
i51
è
n
a fi
i51
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663
n
a
• Standard deviation based on a sample 5 s 5
The Normal
Distribution
fi (xi 2 x) 2
i51
n
è
a a fi b 2 1
i51
A normal curve represents large sets of data that occur naturally, such as
heights, weights, or test grades. The normal curve is bell shaped, symmetric with
respect to a vertical line through its highest point, which is at the mean.
A normal distribution is a set of data that can be represented by a normal
curve. For a normal distribution, the following relationships exist.
1. The mean and the median of the data values lie on the line of symmetry of
the curve.
2. Approximately 68% of the data values lie within one standard deviation
from the mean.
3. Approximately 95% of the data values lie within two standard deviations
from the mean.
4. Approximately 99.7% of the data values lie within three standard deviations from the mean.
The z-score for a data value is the deviation from the mean divided by the
standard deviation. Let x be a data value of a normal distribution.
x2x
x2x
z-score 5 standard
deviation 5 s
The z-score of x, a value from a normal distribution, is positive if x is above
the mean and negative if x is below the mean. The z-score tells us how many
standard deviations x is above or below the mean.
Bivariate Data
A scatter plot can help us to observe the relationship between elements of
the pairs of bivariate data. The correlation between two variables is positive
when the values of the second element of the pairs increase when the values of
the first elements of the pairs increase. The correlation is negative when the values of the second element of the pairs decrease when the values of the first elements of the pairs increase.
The correlation coefficient is a number that indicates the strength and direction of the linear correlation between variables in a set of bivariate data. The
correlation coefficient, r, is a number between 21 and 1. When the absolute
value of the correlation coefficient is close to 1, the data has a strong linear correlation. When the absolute value of the correlation coefficient is close to 0,
there is weak or no linear correlation.
Not all bivariate data can be represented by a linear function. Some data
can be better approximated by a curve. By graphing data in a scatter plot, expo-
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nential, logarithmic, or power regressions may be identified. A calculator can
also be used to find specific linear, exponential, logarithmic, or power functions
that best represent the data.
The process of finding a function value between given values is called interpolation. The process of using pairs of values within a given range to approximate values outside of the given range of values is called extrapolation.
A function value can be found by substituting in the equation of the line of
best fit. Extrapolation can sometimes lead to inaccurate results when the
extreme values of the data do not fit the pattern of the rest of the data.
VOCABULARY
15-1
Statistics • Data • Univariate statistics • Census • Survey • Controlled
experiment • Observational study • Population • Sample • Stem-andleaf diagram • Stem • Leaf • Frequency distribution table • Histogram
15-2
Measure of central tendency • Mean • Arithmetic mean • Median •
Mode • Quartiles • First quartile • Lower quartile • Second quartile •
Third quartile • Upper quartile • Five statistical summary • Box-andwhisker plot
15-3
Cumulative frequency • Percentile
15-4
Measure of dispersion • Range • Interquartile range • Outlier
15-5
Variance • Standard deviation • Standard deviation for a population
(s) • Standard deviation for a sample (s)
15-6
Normal curve • Normal distribution • z-score
15-7
Bivariate statistics • Scatter plot • Correlation • Regression line • Line
of best fit
15-8
Correlation coefficient (r)
15-9
Sinusoidal
15-10 Interpolation • Extrapolation
REVIEW EXERCISES
In 1–3, determine if the data to be collected is univariate or bivariate.
1. The ages of the 20 members of a book club
2. The median heights of boys for each year from age 12 to 18
3. The weight and weight-loss goals of the members of an exercise program
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4. Name and describe four common ways of obtaining data for a statistical
study.
5. In order to determine the average grade for all students who took a test
given to all 9th-grade students in the state, a statistics student at a local
college gathered the test grades for five randomly chosen students in the
high school of the town where he lived.
a. Do the data represent the population or a sample? Explain your
answer.
b. Can the person collecting the data expect that the data collected will
reflect the grades of all students who took the test? Explain why or why
not.
6. Sue’s grades are 88, 87, 85, 82, 80, 80, 78, and 60.
a. What is the range of her grades?
b. What is the mean grade?
c. What is the median grade?
d. What are the upper and lower quartiles?
e. What is the interquartile range?
f. Is the grade of 60 an outlier? Justify your answer.
g. Draw a box-and-whisker plot for Sue’s grades.
7. The hours, xi, that Peg worked for each of the
last 15 weeks are shown in the table at the
right. Show your work. In a through h, you
are not allowed to use the STAT menu of the
calculator.
Hours
xi
Frequency
fi
42
1
40
2
a. Find the mean.
39
2
b. Find the median.
38
5
c. Find the mode.
37
3
d. What is the range?
36
2
e. What are the first and third quartiles?
f. What is the interquartile range?
g. Find the variance.
h. Find the standard deviation.
i. Use the STAT menu on a calculator and compare the values given with
those found in a, b, e, and h.
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8. Each time Kurt swims, he times a few random laps. His times, in seconds,
are shown in the following table.
Time
79
80
81
82
83
84
85
86
87
88
89
90
Frequency
1
3
9
17
30
40
41
29
18
8
3
1
a. What percent of the data lie within 1 standard deviation from the mean?
b. What percent of the data lie within 2 standard deviations from the mean?
c. Does the data appear to represent a normal distribution? Justify your
answer.
In 9–12, for each of the given scatter plots: a. Describe the correlation as strong
positive linear correlation, moderate positive linear correlation, no linear correlation, moderate negative linear correlation, or strong negative linear correlation. b. Determine whether the correlation coefficient would be positive or
negative.
9. 80
70
60
50
40
30
20
10
10. 8
7
6
5
4
3
2
1
1
2
3 4
5
6
7
2 4
11. 400
12.
300
200
100
10
20
30
40
6
8 10 12 14 16
400
350
300
250
200
150
100
50
25
50
75
100 125
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Review Exercises
13. A school investment club chose nine stocks at the beginning of the year
that it believed would represent a good investment portfolio. The chart
shows the price of the stock at the beginning of the year and the gain (a
positive number) or loss (a negative number) at the end of the year.
Stock
Price
Gain
or Loss
31.94
7.19
18.12
20.76 29.65
16.18
10.14 45.85
56.30
40.15
4.20 23.57 22.61 23.32 25.75 14.76
9.23
23.93
a. Draw a scatter plot for the data.
b. Does there appear to be a linear correlation between the stock price and
the gain or loss? If so, is the correlation strong or moderate?
In 14 and 15, find: a. the equation of the linear regression model that appears to
be appropriate for the data. b. the value of the correlation coefficient.
14. The table lists the six states that had the largest percent of increase in population from 2006 to 2007. Population is given in millions, that is, 2.49 represents 2,490,000.
State
Nevada
Arizona
Utah
Idaho Georgia
N. Carolina
Population 2006
2.49
6.17
2.58
1.46
9.34
8.87
Population 2007
2.57
6.34
2.65
1.50
9.54
9.06
15. The salaries that Aaron earned for four years are shown in the table. The
salary is given to the nearest thousand dollars.
Year
1
2
3
4
Salary
52
54
61
63
In 16 and 17, determine the equation of the non-linear regression model that
appears to be appropriate for the data.
16. Mrs. Brudek bakes and sells cookies. The table shows the number of
dozens of cookies that she baked each week for the first seven weeks of
this year.
Week
Dozens of Cookies
1
2
3
4
5
6
7
120 139 162 191 222 257 300
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17. In a local park, an attempt is being made to control the deer population.
The table shows the estimated number of deer in the park for the six years
that the program has been in place.
Year
1
Number of Deer
2
3
4
5
6
700 525 425 350 300 250
In 18–21, use your answers to Exercises 14–17 to estimate the required values.
18. Use the information in Exercise 14 to find the approximate population of
Texas in 2007 if the population in 2006 was 23.9 million.
19. Use the information in Exercise 15 to predict Aaron’s salary in the 5th year.
20. If the number of dozens of cookies that Mrs. Brudek bakes continues to
increase according to the pattern shown in Exercise 16, how many dozen
cookies will she bake in week 8?
21. a. Use the information in Exercise 17 to find the estimated number of
deer in the 8th year of the program if the number of deer continues to
decrease according to the given pattern.
b. Using the information shown in Exercise 17, in what year will the number of deer be approximately 200?
Exploration
The graphing calculator fits a non-linear function to a set of data by transforming the data in such a way that the resulting data fall into a linear pattern. The
calculator then fits a linear equation to the transformed data and applies an
inverse function to find the equation of the non-linear function. In this
Exploration, we will be using this procedure.
Look at the following set of data.
x
–2
21
1
2
3
4
5
6
7
8
y
13
20
29
45
65
114
188
258
378
583
2.058
2.273
2.412
2.578
2.766
ln y
1.125 1.294 1.504
1.699 1.830
The scatter plot on the left reveals that the data is exponential. The scatter
plot on the right is a graph of the ordered pairs (x, y9) where y9 5 ln y. Notice
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that this scatter plot indicates a clear linear pattern. The data is said to have
been linearized. If we now find the linear regression equation of the ordered
pairs (x, y9), the equation is
y9 5 0.387x 1 3.197
However, this equation is really
ln y 5 0.387x 1 3.197
To find the exponential regression model, raise
both sides to the power e.
ln y 5 0.387x 1 3.197
eln y 5 e0.387x 1 3.197
y 5 e0.387x ? e3.197
y 5 (e0.387) x(e3.197)
y 5 24.459(1.473) x
The scatter plot on the right shows the original data and the graph of the exponential regression equation.
For other types of non-linear data, we can follow a similar procedure. First
we linearize the data using the appropriate substitution(s). Then we find the linear regression equation on the transformed data. Lastly, we undo the transformation to obtain the non-linear regression equation.
The substitutions to linearize power and logarithmic data are given below:
Power:
(x, y) → (ln x, ln y) to obtain ln y 5 a ln x 1 b
Logarithmic: (x, y) → (ln x, y)
to obtain
y 5 a ln x 1 b
1. Find a power model using the techniques of this Exploration for the data
given in Example 1 of Section 15-9.
2. Find a logarithmic model using the techniques of this Exploration for the
data given in Example 2 of Section 15-9.
CUMULATIVE REVIEW
CHAPTERS 1–15
Part I
Answer all questions in this part. Each correct answer will receive 2 credits. No
partial credit will be allowed.
1. Solve for x: 2 1 !x 1 3 5 6
(1) 1
(2) 13
(3) 31
(4) 33
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2. An angle measure of 240° is equivalent to
(1) p radians
(3) 43p radians
(4) 23p radians
(2) 34p radians
3. If f(x) 5 3x 1 2 and g(x) 5 x2 2 2, then f(g(3)) is equal to
(1) 18
(2) 23
(3) 77
(4) 119
4. In the set of real numbers, what is the largest possible domain of the func2 2 2
tion f(x) 5 2x
x 2 1 ?
(1) {x : x 1}
(3) {x : x 2}
(2) {x : x 1 and x 21}
(4) {x : x is any real number}
!236
5. The expression 2!36
is equal to
(1) 1
(2) 21
(3) i
(4) 2i
6. If sin u , 0 and tan u 5 245, in what quadrant does the terminal side of u
lie when u is in standard position?
(1) I
(2) II
(3) III
(4) IV
a21
7. For a 0, 1, 21, the expression a2 a2 1 is equivalent to
a2
a
(1) a 1
1
1
(2) a 1
a
a
(3) a 2
1
1
(4) a 2
a
8. The roots of the equation x2 1 7x 2 8 5 0 are
(1) real, rational, and equal
(3) real, irrational, and unequal
(2) real, rational, and unequal
(4) imaginary
9. The sum of the infinite geometric series when the first term is 8 and the
common ratio is 0.2 is
(1) 0.1
(2) 1
(3) 10
(4) 40
10. The product (22 1 6i)(3 1 4i) is equal to
(1) 26 1 24i
(3) 18 1 10i
(2) 26 2 24i
(4) 230 1 10i
Part II
Answer all questions in this part. Each correct answer will receive 2 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
11. What are the roots of x2 2 6x 1 13 5 0?
12. Sketch the graph of y 5 2 cos 12x in the interval 22p x 2p.
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Part III
Answer all questions in this part. Each correct answer will receive 4 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
3
13. If log 2 5 a and log 3 5 b, express log !96 in terms of a and b.
14. Solve for x: 27x 1 1 5 81x
Part IV
Answer all questions in this part. Each correct answer will receive 6 credits.
Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.
15. A tree service wants to estimate the
height of a tree. At point A, the angle
of elevation of the top of the tree is
50°. From point B, 20 feet farther
from the foot of the tree than point A,
the angle of elevation of the top of
the tree is 40°. If points A and B lie
on a line perpendicular to the tree
trunk at its base, what is the height of
the tree to the nearest foot?
16. Solve the following system of equations graphically:
y 5 2x2 1 2x
y 5 2x 2 1
50°
40°
A 20 ft
B
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CHAPTER
16
CHAPTER
TABLE OF CONTENTS
16-1 The Counting Principle
16-2 Permutations and
Combinations
16-3 Probability
16-4 Probability with Two
Outcomes
16-5 Binomial Probability and the
Normal Curve
16-6 The Binomial Theorem
Chapter Summary
Vocabulary
Chapter Review
Cumulative Review
672
PROBABILITY
AND THE
BINOMIAL
THEOREM
Medical facilities and pharmaceutical companies
engage in research to develop new ways to cure or
prevent disease and to ease pain. Often after years of
experimentation a new drug or therapeutic technique
will be developed. But before this new drug or technique can be accepted for general use, it is necessary
to determine its effectiveness and the possible side
effects by studying data from an experimental group of
people who use the new discovery. The principles of
statistics provide the guidelines for choosing such a
group, for conducting the necessary study, and for evaluating the results.
This chapter will review some of the basic principles of probability and extend these principles to questions such as those above.
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16-1 THE COUNTING PRINCIPLE
Tossing a coin is familiar to everyone. We say that when a fair coin is tossed,
either side, heads or tails, is equally likely to appear face up. If a penny and a
nickel are tossed, each coin can show heads, H, or tails, T. Each result is called
an outcome. If we designate the outcomes on the penny as H and T and the outcomes on the nickel as H and T, then the set of possible outcomes for tossing the
two coins contains four pairs:
{(H, H), (H, T), (T, H), (T, T)}
6
This set of all possible outcomes is the sample space for the activity.
If a nickel and a die are tossed, there are 2 ways in which the nickel can land
and 6 ways in which the die can land. There are 2 6 or 12 possible outcomes.
The sample space is shown as a graph of ordered pairs to the left and as a set of
ordered pairs below:
5
4
3
{(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6), (H, 1), (H, 2),
(H, 3), (H, 4), (H, 5), (H, 6)}
2
1
These examples illustrate the Counting Principle.
H
T
The Counting Principle: If one activity can occur in any of m ways and a
second activity can occur in any of n ways, then both activities can occur in
the order given in m n ways.
The counting principle can be applied to any number of activities. For
example, consider a set of three cards lying facedown numbered 4, 6, and 8.
Cards are drawn, one at a time, and not replaced. The numbers are used in the
order in which they are drawn to form three-digit numbers. Note that since
each draw comes from the same set of cards and the card or cards previously
drawn are not replaced, the number of cards available decreases by 1 after
each draw.
The table below represents the number of ways the cards may be drawn.
However, it does not show us what those cards may be. We can draw a tree
diagram to represent the outcomes.
Draw
Number of Cards
Number of Outcomes
1st
3
3
2nd
2
33256
3rd
1
3323156
8
6
4
6
4
4
6
8
4
4
8
8
6
6
8
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Probability and the Binomial Theorem
• There are 6 outcomes in the sample space: {468, 486, 684, 648, 846, 864}.
• The outcomes greater than 800 are the elements of the subset {846, 864}.
• The outcomes less than 650 are the elements of the subset {468, 486, 648}.
• The outcomes that are odd numbers are the elements of the subset { } or
, the empty set.
• The outcomes that are even numbers are the elements of the subset
{468, 486, 684, 648, 846, 864}, that is, the sample space.
A subset of the sample space is an event. In the example above, cards were
drawn without replacement. The outcome of each draw depended on the previous, so the draws are dependent events.
Consider a similar example. There are three cards lying facedown numbered
4, 6, and 8. Three cards are drawn, one at a time, and replaced after each draw.
The numbers are used in the order in which they are drawn to form three-digit
numbers. Note that since each card previously drawn is replaced, the number of
cards available is the same for each draw.
Draw
Number of Cards
Number of Outcomes
1st
3
3
2nd
3
33359
3rd
3
3 3 3 3 3 5 27
4
4
6
6
8
4
6
8
8
4
6
8
4 6 8 4 6 8 4 6 8 4 6 8 4 6 8 4 6 8 4 6 8 4 6 8 4 6 8
The number of possible outcomes when the drawn card is replaced and can
be drawn again is significantly greater than the number of outcomes in the similar example without replacement. Since the outcome of each draw does not
depend on the previous, we call the draws independent events.
EXAMPLE 1
In a debate club, there are 12 boys and 15 girls.
a. In how many ways can a boy and a girl be selected to represent the club at
a meeting?
b. Are the selection of a boy and the selection of a girl dependent or independent events?
Solution a. Use the counting principle. There are 12 possible boys and 15 possible girls
who can be selected. There are 12 3 15 or 180 possible pairs of a boy and a
girl that can be selected.
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b. The selection of a boy does not affect the selection of a girl, and vice versa.
Therefore, these are independent events.
EXAMPLE 2
The skating club is holding its annual competition at which a gold, a silver, and
a bronze medal are awarded.
a. In how many different ways can the medals be awarded if 12 skaters are
participating in the competition?
b. Are the awarding of medals dependent or independent events?
Solution a. • The gold medal can be awarded to one of the 12 skaters.
• After the gold medal is awarded, the silver medal can be awarded to one
of 11 skaters.
• After the silver medal is awarded, the bronze medal can be awarded to
one of 10 skaters.
There are 12 3 11 3 10 or 1,320 possible ways the medals can be awarded.
b. Since the silver medalist cannot have won the gold and the bronze medalist
cannot have won either the gold or silver, the awarding of medals are
dependent events.
Exercises
Writing About Mathematics
1. How is choosing a boy and a girl from 12 boys and 12 girls to represent a club different
from choosing two girls from 12 girls to be president and treasurer of the club?
2. Compare the number of ordered pairs of cards that can be drawn from a deck of 52 cards
without replacement to the number of ordered pairs of cards that can be drawn from a deck
of 52 cards with replacement.
Developing Skills
In 3–6, for the given values of r and n, find the number of ordered selections of r objects from a collection of n objects without replacement.
3. r 5 4, n 5 4
4. r 5 4, n 5 5
5. r 5 3, n 5 8
6. r 5 2, n 5 12
In 7–10, for the given values of r and n, find the number of ordered selections of r objects from a
collection of n objects with replacement.
7. r 5 4, n 5 4
8. r 5 4, n 5 5
9. r 5 2, n 5 8
10. r 5 3, n 5 5
11. For the sample space {A, B, C, D}, determine how many events are possible.
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Probability and the Binomial Theorem
In 12–17, state whether the events are independent or dependent.
12. Tossing a coin and rolling a die
13. Selecting a winner and a runner-up from 7 contestants
14. Picking 2 cards from a standard deck without replacing the first card
15. Buying a magazine and a snack for a train trip
16. Choosing a size and a color for a sweater
17. Writing a 5-letter word using the alphabet if letters cannot be repeated
In 18–27, determine the number of possible outcomes.
18. Tossing a die 3 times
19. Tossing a coin 5 times
20. Choosing breakfast of juice, cereal, and fruit from 3 juices, 5 cereals, and 4 fruits
21. Assembling an outfit from 6 shirts, 4 pairs of pants, and 2 pairs of shoes
22. Choosing a 4-digit entry code using 0–9 if digits cannot be repeated
23. Choosing from 5 flavors of iced tea in 3 different sizes with or without sugar
24. Ordering an ice cream cone from a choice of 31 flavors, 3 types of cone, with or without one
of 4 toppings
25. Making a 7-character license plate using the letters of the alphabet and the digits 1–4 if the
first three characters must be non-repeating letters and the remaining four are digits that
may repeat
26. Seating Andy, Brenda, Carlos, Dabeed, and Eileen in a row of 5 seats
27. Seating Andy, Brenda, Carlos, Dabeed, and Eileen in a row of 5 seats with Andy and Eileen
occupying end seats
Applying Skills
28. In how many ways can 5 new bus passengers choose seats if there are 8 empty seats?
29. In how many ways can 3 different job openings be filled if there are 8 applicants?
30. In how many ways can 5 roles in the school play be filled if there are 9 possible people trying out? (Each role is to be played by a different person.)
31. From the set {2, 3, 4, 5, 6}, a number is drawn and replaced. Then a second number is drawn.
How many two-digit numbers can be formed?
32. In a class of 22 students, the teacher calls on a student to give the answer to the first homework
problem and then calls on a student to give the answer to the second homework problem.
a. How many possible choices could the teacher have made if the same student was not
called on twice?
b. How many possible choices could the teacher have made if the same student may have
been called on twice?
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33. A manufacturer produces jeans in 9 sizes, 7 different shades of blue, and 6 different leg
widths. If a branch store manager orders two pairs of each possible type, how many pairs of
jeans will be in stock?
34. A restaurant offers a special in which a diner can have a choice of appetizer, entree, and
dessert for $11.99.
Appetizer
Mozzarella sticks
Garden salad
Tomato soup
Entree
Dessert
Roast chicken
Steamed fish
Roast beef
Vegetable curry
Chocolate pie
Apple pie
Cherry pie
Blueberry pie
Strawberry-rhubarb pie
Lemon meringue pie
a. How many different meals can a diner order?
b. How many of these choices contain vegetable curry?
c. If a diner does not want to eat the soup or the fish, how many different meals can she
order?
35. On a certain ski run, there are 8 places where a skier can choose to go to the left or to the
right. In how many different ways can the skier cover the run?
36. Stan has letter tiles A M T E .
a. How many different ways can Stan arrange all four tiles?
b. How many different arrangements begin with a vowel?
37. Six students will be seated in a row in the classroom.
a. How many different ways can they be seated?
b. If one student forgot his eyeglasses and must occupy the front seat, how many different
seatings are possible?
38. To duplicate a key, a locksmith begins with a dummy key that has several sections. The locksmith grinds a specific pattern into each section.
a. A particular brand of house key includes 6 sections, and there are 4 possible patterns
for each section. How many different house keys are possible?
b. A desk key has 3 sections, and 64 different keys are possible. How many patterns
are available for each section if each section has the same number of possible
patterns?
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Probability and the Binomial Theorem
39. A physicist, a chemist, a biologist, an astronomer, a geologist, and a mathematician are guest
speakers at a government-sponsored forum on scientific research in the twenty-first century.
The speakers will be seated in a row on a raised platform at the front of the meeting room.
a. How many different ways can the speakers be seated?
b. The astronomer and the mathematician have co-written a paper that they will present.
How many ways can the speakers be seated if the astronomer and the mathematician
wish to sit side-by-side?
40. A given region has the telephone prefix 472.
a. How many 7-digit telephone numbers are possible with this prefix?
b. How ma
